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1 Generalized coordinates 11.1 Constraints and degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Non-holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Physical quantities in generalized coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Generalized momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Double pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Generalized coordinates and virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Lagrangian mechanics 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Definition of the Lagrangian (non-relativistic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Lagrange’s equations of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Lagrange’s equations of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 From Newtonian to Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 Newton’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 D'Alembert’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.3 Equations of motion from D'Alembert’s principle . . . . . . . . . . . . . . . . . . . . . . 142.4.4 Euler–Lagrange equations and Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . 142.4.5 Lagrange multipliers and constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Properties of the Euler–Lagrange equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.1 Non uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.2 Invariance under point transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.3 Cyclic coordinates and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.4 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.5 Mechanical similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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2.5.6 Interacting particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Examples in specific coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6.2 Polar coordinates in 2d and 3d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Non-relativistic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7.1 Pendulum on a movable support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7.2 Two-body central force problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7.3 Non-relativistic test particles in fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Extensions to include non-conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.9 Definitions of the Lagrangian (relativistic mechanics) . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9.1 Coordinate formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.9.2 Covariant formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.10 Relativistic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.10.1 Special relativistic 1d harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.10.2 Special relativistic constant force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.10.3 Special relativistic test particle in an electromagnetic field . . . . . . . . . . . . . . . . . . 242.10.4 General relativistic test particle in an electromagnetic field . . . . . . . . . . . . . . . . . . 24
2.11 Applications or extensions of Lagrangian mechanics in other contexts . . . . . . . . . . . . . . . . 252.11.1 Relation to other formulations of classical mechanics . . . . . . . . . . . . . . . . . . . . . 252.11.2 Applications in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.11.3 Classical field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.11.4 Uses in Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.13 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.16 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Action (physics) 303.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Mathematical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Action in classical physics (disambiguation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Action (functional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.2 Abbreviated action (functional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 Hamilton’s principal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.4 Hamilton’s characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.5 Other solutions of Hamilton–Jacobi equations . . . . . . . . . . . . . . . . . . . . . . . . 313.4.6 Action of a generalized coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.7 Action for a Hamiltonian flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Euler–Lagrange equations for the action integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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3.5.1 Example: free particle in polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 The action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6.1 Classical fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.3 Quantum mechanics and quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . 333.6.4 Single relativistic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.5 Modern extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.9 Sources and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 AQUAL 354.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Averaged Lagrangian 365.1 Resulting equations for pure wave motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.2 Slowly-varying waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.3 Averaged Lagrangian for slowly-varying waves . . . . . . . . . . . . . . . . . . . . . . . . 385.2.4 Set of equations emerging from the averaged Lagrangian . . . . . . . . . . . . . . . . . . . 385.2.5 Mean motion and pseudo-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Conservation of wave action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 Conservation of energy and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 Connection to the dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6.2 Publications by Whitham on the method . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.6.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Canonical coordinates 426.1 Definition, in classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Definition, on cotangent bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 Formal development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4 Generalized coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Classical field theory 447.1 Non-relativistic field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.1.1 Newtonian gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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7.1.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.1.3 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2 Relativistic field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2.1 Lagrangian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.3 Relativistic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.3.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.3.2 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8 Covariant classical field theory 488.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9 D'Alembert’s principle 499.1 General case with changing masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.2 Derivation for special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.3 D'Alembert’s principle of inertial forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9.3.1 Example for 1D motion of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.3.2 Example for plane 2D motion of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . 519.3.3 Dynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.4 References[8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10 Fiber derivative 53
11 FLEXPART 5411.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
12 Generalized forces 5512.1 Virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
12.1.1 Generalized coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.1.2 Generalized forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.1.3 Velocity formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
12.2 D'Alembert’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
13 Geometric mechanics 5713.1 Momentum map and reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.2 Variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.3 Geometric integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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13.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14 Gibbons–Hawking–York boundary term 5914.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
15 Hamilton’s principle 6015.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
15.1.1 Euler–Lagrange equations derived from the action integral . . . . . . . . . . . . . . . . . . 6015.1.2 Canonical momenta and constants of motion . . . . . . . . . . . . . . . . . . . . . . . . . 6115.1.3 Example: Free particle in polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 61
15.2 Hamilton’s principle applied to deformable bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.3 Comparison with Maupertuis’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.4 Action principle for fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
15.4.1 Classical field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.4.2 Quantum mechanics and quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . 62
15.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
16 Inverse problem for Lagrangian mechanics 6416.1 Background and statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.2 Douglas’ theorem and the Helmholtz conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
16.2.1 Applying Douglas’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
17 Jacobi coordinates 6617.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
18 Joseph-Louis Lagrange 6818.1 Scientific contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.2 Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
18.2.1 Early years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.2.2 Berlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6918.2.3 Paris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
18.3 Work in Berlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.3.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.3.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.3.3 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.3.4 Other mathematical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.3.5 Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.3.6 Mécanique analytique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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18.4 Work in France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.4.1 Differential calculus and calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . 7318.4.2 Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.4.3 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.4.4 Celestial mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
18.5 Prizes and distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
19 Lagrangian point 7619.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7619.2 Lagrange points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7619.3 Natural objects at Lagrangian points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7719.4 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
19.4.1 L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.4.2 L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.4.3 L3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.4.4 L4 and L5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
19.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.6 Spaceflight applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
19.6.1 Spacecraft at Sun–Earth L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.6.2 Spacecraft at Sun–Earth L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.6.3 List of missions to Lagrangian points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
19.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
20 Lagrangian system 8320.1 Lagrangians and Euler–Lagrange operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
20.1.1 In coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.1.2 Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
20.2 Cohomology and Noether’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.3 Graded manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.4 Alternative formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.5 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8420.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8420.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8420.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
CONTENTS vii
21 Minimal coupling 8521.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
22 Monogenic system 8622.1 Mathematical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
23 Ostrogradsky instability 8723.1 Outline of proof [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
24 Palatini variation 8824.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8824.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
25 Rayleigh dissipation function 8925.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
26 Rheonomous 9026.1 Example: simple 2D pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
27 Scleronomous 9227.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.2 Example: pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
28 Tautological one-form 9428.1 Coordinate-free definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.4 On metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
29 Total derivative 9629.1 Differentiation with indirect dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9629.2 The total derivative via differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.3 The total derivative as a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
viii CONTENTS
29.4 Total differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.5 Application to equation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
30 Virtual displacement 9930.1 Comparison between virtual and actual displacements . . . . . . . . . . . . . . . . . . . . . . . . 9930.2 Virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 101
30.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10130.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 1
Generalized coordinates
In analytical mechanics, specifically the study of the rigidbody dynamics of multibody systems, the term general-ized coordinates refers to the parameters that describethe configuration of the system relative to some referenceconfiguration. These parameters must uniquely definethe configuration of the system relative to the referenceconfiguration.[1] The generalized velocities are the timederivatives of the generalized coordinates of the system.An example of a generalized coordinate is the angle thatlocates a point moving on a circle. The adjective “gen-eralized” distinguishes these parameters from the tradi-tional use of the term coordinate to refer to Cartesiancoordinates: for example, describing the location of thepoint on the circle using x and y coordinates.Although there may be many choices for generalized co-ordinates for a physical system, parameters which areconvenient are usually selected for the specification ofthe configuration of the system and which make the solu-tion of its equations of motion easier. If these parametersare independent of one another, the number of indepen-dent generalized coordinates is defined by the number ofdegrees of freedom of the system.[2] [3]
1.1 Constraints and degrees offreedom
Open straight path
Open curved path F(x, y) = 0
Closed curved path C(x, y) = 0One generalized coordinate, one degree of freedom, onpaths in 2d. Only one number is needed to uniquelyspecify positions on the curve, the examples shownare the arc length s or angle θ. Both of the Cartesiancoordinates (x, y) are unnecessary since either x or y isrelated to the other by the equations of the curves. Theycan also be parameterized by s or θ.
Open curved path F(x, y) = 0
1
2 CHAPTER 1. GENERALIZED COORDINATES
Closed curved path C(x, y) = 0The arc length s along the curve is a legitimate gen-eralized coordinate since the position is uniquelydetermined, but the angle θ is not since there are multi-ple positions owing to the self-intersections of the curves.
Generalized coordinates are usually selected to providethe minimum number of independent coordinates thatdefine the configuration of a system, which simplifies theformulation of Lagrange’s equations of motion. How-ever, it can also occur that a useful set of generalized co-ordinates may be dependent, which means that they arerelated by one or more constraint equations.
1.1.1 Holonomic constraints
q x
y
z C(x(t),y(t),z(t)) = 0
x
y
z S(x,y,z) = 0
q 1 q
2
x
y
z
F(x,y,z) = 0 q
1
q 2
x
y
z
q
F(x(t),y(t),z(t)) = 0
Top: one degree of freedom, bottom: two degrees of freedom,left: an open curve F (parameterized by t) and surface F, right: aclosed curveC and closed surface S. The equations shown are theconstraint equations. Generalized coordinates are chosen anddefined with respect to these curves (one per degree of freedom),and simplify the analysis since even complicated curves are de-scribed by the minimum number of coordinates required.
For a system of N particles in 3d real coordinate space,the position vector of each particle can be written as a3-tuple in Cartesian coordinates;
r1 = (x1, y1, z1) , r2 = (x2, y2, z2) , . . . , rN = (xN , yN , zN ) .
Any of the position vectors can be denoted rk where k =1, 2, ..., N labels the particles. A holonomic constraint isa constraint equation of the form for particle k[4][nb 1]
f(rk, t) = 0
which connects all the 3 spatial coordinates of that parti-cle together, so they are not independent. The constraintmay change with time, so time t will appear explicitly in
the constraint equations. At any instant of time, when t isa constant, any one coordinate will be determined fromthe other coordinates, e.g. if xk and zk are given, then sois yk. One constraint equation counts as one constraint.If there are C constraints, each has an equation, so therewill be C constraint equations. There is not necessarilyone constraint equation for each particle, and if there areno constraints on the system then there are no constraintequations.So far, the configuration of the system is defined by 3Nquantities, but C coordinates can be eliminated, one co-ordinate from each constraint equation. The number ofindependent coordinates is n = 3N − C. (In D dimen-sions, the original configuration would need ND coordi-nates, and the reduction by constraints means n = ND −C). It is ideal to use the minimum number of coordinatesneeded to define the configuration of the entire system,while taking advantage of the constraints on the system.These quantities are known as generalized coordinatesin this context, denoted qj(t). It is convenient to collectthem into an n-tuple
q(t) = (q1(t), q2(t), . . . , qn(t))
which is a point in the configuration space of the sys-tem. They are all independent of one other, and eachis a function of time. Geometrically they can be lengthsalong straight lines, or arc lengths along curves, or angles;not necessarily Cartesian coordinates or other standardorthogonal coordinates. There is one for each degree offreedom, so the number of generalized coordinates equalsthe number of degrees of freedom, n. A degree of free-dom corresponds to one quantity that changes the config-uration of the system, for example the angle of a pendu-lum, or the arc length traversed by a bead along a wire.If it is possible to find from the constraints as many inde-pendent variables as there are degrees of freedom, thesecan be used as generalized coordinates[5] The positionvector rk of particle k is a function of all the n gener-alized coordinates and time,[6][7][8][5][nb 2]
rk = rk(q(t), t) ,
and the generalized coordinates can be thought of as pa-rameters associated with the constraint.The corresponding time derivatives of q are thegeneralized velocities,
q =dqdt
= (q1(t), q2(t), . . . , qn(t))
(each dot over a quantity indicates one time derivative).The velocity vector vk is the total derivative of rk withrespect to time
1.3. EXAMPLES 3
vk = rk =drkdt
=
n∑j=1
∂rk∂qj
qj +∂rk∂t
.
and so generally depends on the generalized velocities andcoordinates. Since we are free to specify the initial valuesof the generalized coordinates and velocities separately,the generalized coordinates and velocities can be treatedas independent variables. The generalized coordinates qjand velocities dqj/dt are treated as independent variables.
1.1.2 Non-holonomic constraints
A mechanical system can involve constraints on both thegeneralized coordinates and their derivatives. Constraintsof this type are known as non-holonomic. First-ordernon-holonomic constraints have the form
g(q, q, t) = 0 ,
An example of such a constraint is a rolling wheel orknife-edge that constrains the direction of the velocityvector. Non-holonomic constraints can also involve next-order derivatives such as generalized accelerations.
1.2 Physical quantities in general-ized coordinates
1.2.1 Kinetic energy
The total kinetic energy of the system is the energy of thesystem’s motion, defined as[9]
T =1
2
N∑k=1
mk rk · rk ,
in which · is the dot product. The kinetic energy is a func-tion only of the velocities vk, not the coordinates rk them-selves. By contrast an important observation is[10]
rk·rk =n∑
i,j=1
(∂rk∂qi
· ∂rk∂qj
)qiqj+
n∑i=1
(2∂rk∂qi
· ∂rk∂t
)qi+
(∂rk∂t
· ∂rk∂t
),
which illustrates the kinetic energy is in general a functionof the generalized velocities, coordinates, and time if theconstraint also varies with time, so T = T(q, dq/dt, t).In the case the constraint on the particle is time-independent, then all partial derivatives with respectto time are zero, and the kinetic energy has no time-dependence and is a homogeneous function of degree 2in the generalized velocities;
rk · rk =n∑
i,j=1
(∂rk∂qi
· ∂rk∂qj
)qiqj .
Still for the time-independent case, this expression isequivalent to taking the line element squared of the tra-jectory for particle k,
ds2k = drk · drk =n∑
i,j=1
(∂rk∂qi
· ∂rk∂qj
)dqidqj ,
and dividing by the square differential in time, dt2, toobtain the velocity squared of particle k. Thus for time-independent constraints it is sufficient to know the lineelement to quickly obtain the kinetic energy of particlesand hence the Lagrangian.[11]
It is instructive to see the various cases of polar coordi-nates in 2d and 3d, owing to their frequent appearance.In 2d polar coordinates (r, θ),
(ds
dt
)2
= r2 + r2θ2 ,
in 3d cylindrical coordinates (r, θ, z),
(ds
dt
)2
= r2 + r2θ2 + z2 ,
in 3d spherical coordinates (r, θ, φ),
(ds
dt
)2
= r2 + r2θ2 + r2 sin2 θ φ2 .
1.2.2 Generalized momentum
The generalized momentum “canonically conjugate to” thecoordinate qi is defined by
pi =∂L
∂qi.
If the Lagrangian L does not depend on some coordinateqi, then it follows from the Euler–Lagrange equations thatthe corresponding generalized momentum will be a con-served quantity, because its time derivative is zero so themomentum must be a constant of the motion;
pi =d
dt
∂L
∂qi=
∂L
∂qi= 0 .
4 CHAPTER 1. GENERALIZED COORDINATES
L
mg
mgcos mgsin m
Dynamic model of a simple pendulum.
1.3 Examples
1.3.1 Simple pendulum
The relationship between the use of generalized coor-dinates and Cartesian coordinates to characterize themovement of a mechanical system can be illustratedby considering the constrained dynamics of a simplependulum.[12][13]
A simple pendulum consists of a mass M hanging from apivot point so that it is constrained to move on a circle ofradius L. The position of the mass is defined by the coor-dinate vector r=(x, y) measured in the plane of the circlesuch that y is in the vertical direction. The coordinates xand y are related by the equation of the circle
f(x, y) = x2 + y2 − L2 = 0,
that constrains the movement of M. This equation alsoprovides a constraint on the velocity components,
f(x, y) = 2xx+ 2yy = 0.
Now introduce the parameter θ, that defines the angularposition of M from the vertical direction. It can be usedto define the coordinates x and y, such that
r = (x, y) = (L sin θ,−L cos θ).
The use of θ to define the configuration of this systemavoids the constraint provided by the equation of the cir-cle.Notice that the force of gravity acting on the mass m isformulated in the usual Cartesian coordinates,
F = (0,−mg),
where g is the acceleration of gravity.The virtual work of gravity on the mass m as it followsthe trajectory r is given by
δW = F · δr.
The variation δr can be computed in terms of the coor-dinates x and y, or in terms of the parameter θ,
δr = (δx, δy) = (L cos θ, L sin θ)δθ.
Thus, the virtual work is given by
δW = −mgδy = −mgL sin θδθ.
Notice that the coefficient of δy is the y-component ofthe applied force. In the same way, the coefficient of δθis known as the generalized force along generalized coor-dinate θ, given by
Fθ = −mgL sin θ.
To complete the analysis consider the kinetic energy T ofthe mass, using the velocity,
v = (x, y) = (L cos θ, L sin θ)θ,
so,
T =1
2mv · v = 1
2m(x2 + y2) =
1
2mL2θ2.
Lagrange’s equations for the pendulum in terms of thecoordinates x and y are given by,
d
dt
∂T
∂x−∂T
∂x= Fx+λ
∂f
∂x,
d
dt
∂T
∂y−∂T
∂y= Fy+λ
∂f
∂y.
This yields the three equations
mx = λ(2x), my = −mg+λ(2y), x2+y2−L2 = 0,
1.3. EXAMPLES 5
in the three unknowns, x, y and λ.Using the parameter θ, Lagrange’s equations take theform
d
dt
∂T
∂θ− ∂T
∂θ= Fθ,
which becomes,
mL2θ = −mgL sin θ,
or
θ +g
Lsin θ = 0.
This formulation yields one equation because there is asingle parameter and no constraint equation.This shows that the parameter θ is a generalized coordi-nate that can be used in the same way as the Cartesiancoordinates x and y to analyze the pendulum.
1.3.2 Double pendulum
L1
L22
1
m1
m2
A double pendulum
The benefits of generalized coordinates become appar-ent with the analysis of a double pendulum. For the twomassesmᵢ, i=1, 2, let rᵢ=(xᵢ, yᵢ), i=1, 2 define their two tra-jectories. These vectors satisfy the two constraint equa-tions,
f1(x1, y1, x2, y2) = r1·r1−L21 = 0, f2(x1, y1, x2, y2) = (r2−r1)·(r2−r1)−L2
2 = 0.
The formulation of Lagrange’s equations for this systemyields six equations in the four Cartesian coordinates xᵢ,yᵢ i=1, 2 and the two Lagrange multipliers λᵢ, i=1, 2 thatarise from the two constraint equations.Now introduce the generalized coordinates θᵢ i=1,2 thatdefine the angular position of each mass of the doublependulum from the vertical direction. In this case, wehave
r1 = (L1 sin θ1,−L1 cos θ1), r2 = (L1 sin θ1,−L1 cos θ1)+(L2 sin θ2,−L2 cos θ2).
The force of gravity acting on the masses is given by,
F1 = (0,−m1g), F2 = (0,−m2g)
where g is the acceleration of gravity. Therefore, the vir-tual work of gravity on the two masses as they follow thetrajectories rᵢ, i=1,2 is given by
δW = F1 · δr1 + F2 · δr2.
The variations δrᵢ i=1, 2 can be computed to be
δr1 = (L1 cos θ1, L1 sin θ1)δθ1, δr2 = (L1 cos θ1, L1 sin θ1)δθ1+(L2 cos θ2, L2 sin θ2)δθ2Thus, the virtual work is given by
δW = −(m1 +m2)gL1 sin θ1δθ1 −m2gL2 sin θ2δθ2,
and the generalized forces are
Fθ1 = −(m1+m2)gL1 sin θ1, Fθ2 = −m2gL2 sin θ2.
Compute the kinetic energy of this system to be
T =1
2m1v1·v1+
1
2m2v2·v2 =
1
2(m1+m2)L
21θ
21+
1
2m2L
22θ
22+m2L1L2 cos(θ2−θ1)θ1θ2.
Lagrange’s equations yield two equations in the unknowngeneralized coordinates θᵢ i=1, 2, given by[14]
(m1+m2)L21θ1+m2L1L2θ2 cos(θ2−θ1)+m2L1L2θ2
2 sin(θ1−θ2) = −(m1+m2)gL1 sin θ1,
and
m2L22θ2+m2L1L2θ1 cos(θ2−θ1)+m2L1L2θ1
2 sin(θ2−θ1) = −m2gL2 sin θ2.
The use of the generalized coordinates θᵢ i=1, 2 providesan alternative to the Cartesian formulation of the dynam-ics of the double pendulum.
6 CHAPTER 1. GENERALIZED COORDINATES
1.4 Generalized coordinates andvirtual work
The principle of virtual work states that if a system is instatic equilibrium, the virtual work of the applied forcesis zero for all virtual movements of the system from thisstate, that is, δW=0 for any variation δr.[15] When formu-lated in terms of generalized coordinates, this is equiva-lent to the requirement that the generalized forces for anyvirtual displacement are zero, that is F ᵢ=0.Let the forces on the system beF , j=1, ..., m be applied topoints with Cartesian coordinates r , j=1,..., m, then thevirtual work generated by a virtual displacement from theequilibrium position is given by
δW =m∑j=1
Fj · δrj .
where δr , j=1, ..., m denote the virtual displacements ofeach point in the body.Now assume that each δr depends on the generalized co-ordinates qᵢ, i=1, ..., n, then
δrj =∂rj∂q1
δq1 + . . .+∂rj∂qn
δqn,
and
δW =
m∑j=1
Fj ·∂rj∂q1
δq1+. . .+
m∑j=1
Fj ·∂rj∂qn
δqn.
The n terms
Fi =
m∑j=1
Fj ·∂rj∂qi
, i = 1, . . . , n,
are the generalized forces acting on the system. Kane[16]shows that these generalized forces can also be formulatedin terms of the ratio of time derivatives,
Fi =m∑j=1
Fj ·∂vj∂qi
, i = 1, . . . , n,
where v is the velocity of the point of application of theforce F .In order for the virtual work to be zero for an arbitraryvirtual displacement, each of the generalized forces mustbe zero, that is
δW = 0 ⇒ Fi = 0, i = 1, . . . , n.
1.5 See also• Hamiltonian mechanics
• Virtual work
• Orthogonal coordinates
• Curvilinear coordinates
• Frenet-Serret formulas
• Mass matrix
• Stiffness matrix
• Generalized forces
1.6 Notes[1] Some authors set the constraint equations to a constant for
convenience with some constraint equations (e.g. pendu-lums), others set it to zero. It makes no difference becausethe constant can be subtracted to give zero on one side ofthe equation. Also, in Lagrange’s equations of the firstkind, only the derivatives are needed.
[2] Some authors e.g. Hand & Finch take the form of the po-sition vector for particle k, as shown here, as the conditionfor the constraint on that particle to be holonomic.
1.7 References[1] Jerry H. Ginsberg (2008). "§7.2.1 Selection of general-
ized coordinates”. Engineering dynamics, Volume 10 (3rded.). Cambridge University Press. p. 397. ISBN 0-521-88303-2.
[2] Farid M. L. Amirouche (2006). "§2.4: Generalized co-ordinates”. Fundamentals of multibody dynamics: theoryand applications. Springer. p. 46. ISBN 0-8176-4236-6.
[3] Florian Scheck (2010). "§5.1 Manifolds of generalizedcoordinates”. Mechanics: From Newton’s Laws to Deter-ministic Chaos (5th ed.). Springer. p. 286. ISBN 3-642-05369-6.
[4] Goldstein 1980, p. 12
[5] Kibble & Berkshire 2004, p. 232
[6] Torby 1984, p. 260
[7] Goldstein 1980, p. 13
[8] Hand & Finch 2008, p. 15
[9] Torby 1984, p. 269
[10] Goldstein 1980, p. 25
[11] Landau & Lifshitz 1976, p. 8
[12] Greenwood, Donald T. (1987). Principles of Dynamics(2nd ed.). Prentice Hall. ISBN 0-13-709981-9.
1.7. REFERENCES 7
[13] Richard Fitzpatrick, Newtonian Dynamics, http://farside.ph.utexas.edu/teaching/336k/Newton/Newtonhtml.html.
[14] Eric W. Weisstein, Double Pendulum, science-world.wolfram.com. 2007
[15] Torby, Bruce (1984). “Energy Methods”. Advanced Dy-namics for Engineers. HRW Series in Mechanical Engi-neering. United States of America: CBS College Publish-ing. ISBN 0-03-063366-4.
[16] T. R. Kane and D. A. Levinson, Dynamics: theory andapplications, McGraw-Hill, New York, 1985
Chapter 2
Lagrangian mechanics
Joseph-Louis Lagrange (1736—1813)
Lagrangian mechanics is a reformulation of classicalmechanics, introduced by the Italian-French mathemati-cian and astronomer Joseph-Louis Lagrange in 1788.In Lagrangian mechanics, the trajectory of a systemof particles is derived by solving the Lagrange equa-tions in one of two forms, either the Lagrange equa-tions of the first kind,[1] which treat constraints explic-itly as extra equations, often using Lagrange multipli-ers;[2][3] or theLagrange equations of the second kind,which incorporate the constraints directly by judiciouschoice of generalized coordinates.[1][4] In each case, amathematical function called the Lagrangian is a func-tion of the generalized coordinates, their time derivatives,and time, and contains the information about the dynam-ics of the system.No new physics is introduced by Lagrangian mechanics;it is actually less general than Newtonian mechanics.[5]Newton’s laws can include non-conservative forces likefriction, however they must include constraint forces ex-plicitly and are best suited to Cartesian coordinates. La-grangian mechanics is ideal for systems with conservative
forces and for bypassing constraint forces, and some (notall) non-conservative forces, in any coordinate system.Generalized coordinates can be chosen by convenience,to exploit symmetries in the system or the geometry ofthe constraints, which considerably simplifies describingthe dynamics of the system. Lagrangian mechanics alsoreveals conserved quantities and their symmetries in adirect way, although only as a special case of Noether’stheorem. The theory connects with the principle of sta-tionary action,[6] although Lagrangian mechanics is lessgeneral because it is restricted to equilibrium problems.[7]Also, Lagrangian mechanics can only be applied to sys-tems with holonomic constraints, because the formula-tion does not work for Nonholonomic constraints. Threeexamples[8] are when the constraint equations are nonin-tegrable, when the constraints have inequalities, or withcomplicated non-conservative forces like friction. Non-holonomic constraints require special treatment, and onemay revert to Newtonian mechanics, or use other meth-ods.The Lagrangian formulation of mechanics is importantnot just for its broad applications, but also for its rolein advancing deep understanding of physics. AlthoughLagrange only sought to describe classical mechanics,Hamilton’s principle that can be used to derive the La-grange equation was later recognized to be applicable tomuch of theoretical physics as well. In quantum mechan-ics, action and quantum-mechanical phase are related viaPlanck’s constant, and the principle of stationary actioncan be understood in terms of constructive interferenceof wave functions. If the Lagrangian is invariant undera symmetry, then the resulting equations of motion arealso invariant under that symmetry. This characteristic isvery helpful in showing that theories are consistent witheither special relativity or general relativity. The actionprinciple, and the Lagrangian formalism, are tied closelyto Noether’s theorem, which connects physical conservedquantities to continuous symmetries of a physical sys-tem. Lagrangian mechanics and Noether’s theorem to-gether yield a natural formalism for first quantization byincluding commutators between certain terms of the La-grangian equations of motion for a physical system.Lagrangian mechanics is widely used to solve mechani-cal problems in physics and engineering when Newton’s
8
2.1. INTRODUCTION 9
formulation of classical mechanics is not convenient. La-grange’s equations are also used in optimisation problemsof dynamic systems. In mechanics, Lagrange’s equationsof the second kind are used much more than those of thefirst kind.
2.1 Introduction
f(x,y) = 0 N
m
C
y
x
s
r = (x,y)
Bead constrained to move on a frictionless wire. The wire exertsa reaction force C on the bead to keep it on the wire. The non-constraint forceN in this case is gravity. Notice the initial positionof the wire can lead to different motions.
C
N
r=(x,y)
m
y
x
L
x 2 y 2 + L 2 - = 0 f(x,y) = Simple pendulum. Since the rod is rigid, the position ofthe bob is constrained according to the equation f(x, y) =0, the constraint force C is the tension in the rod. Againthe non-constraint force N in this case is gravity.
L1
L22
1
m1
m2
A double pendulum consists of two simple pendulums at-
tached end to end. There is a constraint for each pendu-lum bob.The strength of Lagrangian mechanics is its ability to
l l
cos
l
sin l
cos l ( ) 1
sin l
m
Spherical pendulum: angles and velocities.
r =r (q ,q ) 1 1 1 5
r =r (q ,q ) 2 2 2 4
r 1
r 2
r 3
N 2
N 3
C 1
f(x,y,z)=0
m 2 m 1
m 3
q 1
q 2 q 3
q 4
q 5
q 6
C 3
C 2
N 1
f(r )=f(x ,y ,z )=0 1 1 1 1
f(r )=f(x ,y ,z )=0 2 2 2 2
f(r )=f(x ,y ,z )=0 3 3 3 3 r =r (q ,q ) 3 3 3 6
Three particles in three spatial dimensions, in a constrainedsurface f(x, y, z)=0. The constraint forces are C and non-constraint forces are N .
handle constrained mechanical systems. The followingexamples motivate the need for the concepts and termi-nology used to handle such systems.For a bead sliding on a frictionless wire subject only togravity in 2d space, the constraint on the bead can bestated in the form f(r) = 0, where the position of the beadcan be written r = (x(s), y(s)), in which s is a parameter,the arc length s along the curve from some point on thewire. Only one coordinate is needed instead of two, be-cause the position of the bead can be parameterized byone number, s, and the constraint equation connects the
10 CHAPTER 2. LAGRANGIAN MECHANICS
two coordinates x and y; either one is determined fromthe other. The constraint force is the reaction force thewire exerts on the bead to keep it on the wire, and thenon-constraint applied force is gravity acting on the bead.Suppose the wire changes its shape with time, by flexing.Then the constraint equation and position of the particleare respectively
f(r, t) = 0 , r = (x(s, t), y(s, t))
which now both depend on time t due to the changingcoordinates as the wire changes its shape. Notice timeappears implicitly via the coordinates and explicitly in theconstraint equations.Another interesting 2d example is the chaotic double pen-dulum, again subject to gravity. The length of one pen-dulum is L1 and the length of the other is L2. Each pen-dulum bob has a constraint equation,
f(r1) = x21+y21−L2
1 = 0 , f(r2) = x22+y22−L2
2 = 0 ,
where the positions of the bobs are
r1 = (x1(θ1), y1(θ1)) , r2 = (x2(θ2), y2(θ2)) ,
and θ1 is the angle of pendulum 1 from some referencedirection, likewise for pendulum 2. Each pendulum canbe described by one coordinate since the constraint equa-tion for each connects the two spatial coordinates.For a 3d example, a spherical pendulum with constantlength l free to swing in any angular direction subject togravity, the constraint on the pendulum bob can be statedin the form
f(r) = x2 + y2 + z2 − l2 = 0 ,
where the position of the pendulum bob can be written
r = (x(θ, ϕ), y(θ, ϕ), z(θ, ϕ)) ,
in which (θ, φ) are the spherical polar angles because thebob moves in the surface of a sphere. A logical choiceof variables to describe the motion are the angles (θ, φ).Notice only two coordinates are needed instead of three,because the position of the bob can be parameterized bytwo numbers, and the constraint equation connects thethree coordinates x, y, z so any one of them is determinedfrom the other two.For analyzing the small oscillations of multiple coupledsimple harmonic oscillators, Lagrangian mechanics is es-pecially natural, since the kinetic and potential energiesof the system take a simple form despite the fact there
are many particles, and the equations of motion can bederived immediately.For N particles in 3d space, the position vector of eachparticle can written as a 3-tuple in Cartesian coordinates
r1 = (x1, y1, z1) , r2 = (x2, y2, z2) , . . . , rN = (xN , yN , zN ) ,
so overall, there are 3N coordinates to define the configu-ration of the system. These are all specific points in spaceto locate the particles, a general point in space is writtenr = (x, y, z). If any or all of the particles are subject toa holonomic constraint, described by a constraint equa-tion of the form f(r, t) = 0, then at any instant of timethe position coordinates of those particles are linked to-gether and not independent. If there are C constraints inthe system, then each has a constraint equation,
f1(r, t) = 0 , f2(r, t) = 0 , . . . , fC(r, t) = 0 ,
and one coordinate can be eliminated from each con-straint equation. The number of independent coordinatesis therefore n = 3N − C. We can transform each posi-tion vector to a common set of n generalized coordinates,conveniently written as an n-tuple q = (q1, q2, ... qn), byexpressing each position vector, and hence the positioncoordinates, as functions of the generalized coordinatesand time,
rk = rk(q, t) = (xk(q, t), yk(q, t), zk(q, t), t) .
The time derivatives of the generalized coordinates arecalled the generalized velocities, and for each particle thetransformation of its velocity vector, the total derivativeof its position with respect to time, is
vk = rk =drkdt
=
(dxk
dt,dykdt
,dzkdt
), qj =
dqjdt
, vk =n∑
j=1
∂rk∂qj
qj+∂rk∂t
,
(each overdot indicates a time derivative).In the previous examples, if one tracks each of the mas-sive objects as a particle (bead, pendulum bob, etc.), cal-culation of the motion of the particle using Newtonianmechanics would require solving for the time-varyingconstraint force required to keep the particle in the con-strained motion (reaction force exerted by the wire onthe bead, or tension in the pendulum rods). For the sameproblem using Lagrangian mechanics, one looks at thepath the particle can take and chooses a convenient setof independent generalized coordinates that completelycharacterize the possible motion of the particle. Thischoice eliminates the need for the constraint force to en-ter into the resultant system of equations. There are fewerequations since one is not directly calculating the influ-ence of the constraint on the particle at a given moment.
2.3. EQUATIONS OF MOTION 11
2.2 Definition of the Lagrangian(non-relativistic)
The central quantity of Lagrangian mechanics is the La-grangian, a function which summarizes the dynamics ofthe entire system. Overall, the Lagrangian has no singleexpression for all physical systems. Any function whichgenerates the correct equations of motion from the Euler–Lagrange equations, in agreement with physical laws, canbe taken as a Lagrangian. However, it is possible toconstruct general expressions for large classes of appli-cations.The non-relativistic Lagrangian for a system of particlescan be defined by[9]
L = T − V ,
where T is the total kinetic energy of the system, and Vthe potential energy of the system. The dimensions of theLagrangian are the same as energy.The total kinetic energy of a system of N particles withmasses m1, m2, ..., mN is the sum (indicated by Σ) of thekinetic energies of the particles, the energy of the system’smotion,[10]
T =1
2
N∑k=1
mkv2k , v2k = vk · vk = rk · rk ,
where · is the dot product. The kinetic energy is a functiononly of the velocities vk, not the positions rk nor time t, soT = T(v1, v2, ...). By contrast, the above expression forvelocity shows the kinetic energy in generalized coordi-nates depends on the generalized velocities, coordinates,and time if the position vectors depend explicitly on timedue to time-varying constraints, so T = T(q, dq/dt, t).The potential energy V of the system reflects the energyof interaction between the particles, i.e. how much en-ergy any one particle will have due to all the others andother external influences. For conservative forces, it is afunction of the position vectors of the particles only, so V= V(r1, r2, ...). For those non-conservative forces whichcan be derived from an appropriate potential, the veloc-ities will appear also, V = V(r1, r2, ..., v1, v2, ...). Ifthere is some external field changing with time, or exter-nal driving force, the potential will change with time, somost generally V = V(r1, r2, ..., v1, v2, ..., t).If the potential or kinetic energy, or both, depend explic-itly on time due to time-varying constraints or externalinfluences, the Lagrangian L(q, dq/dt, t) is explicitly time-dependent. If neither the potential nor the kinetic energydepend on time, then the Lagrangian L(q, dq/dt) is explic-itly independent of time. In either case, the Lagrangianwill always have implicit time-dependence through thegeneralized coordinates.
Even more generally, in addition to the Lagrangian, it issometimes possible to introduce another function to ac-count for dissipative forces. In relativistic mechanics, Lneeds more subtle attention, because it is not the differ-ence between kinetic and potential energy. These casesare detailed later.
2.3 Equations of motion
Although the equations of motion include partial deriva-tives denoted by ∂/∂, they are still ordinary differentialequations in the position coordinates of the particles. Thetotal time derivative denoted d/dt may involve implicitdifferentiation. By solving the equations, subject to theinitial values of the positions and velocities, will give thepositions of the particles as functions of time, and onecan see how the system evolves.
2.3.1 Lagrange’s equations of the firstkind
Lagrange introduced an analytical method for finding sta-tionary points using the method of Lagrange multipliers,and also applied it to mechanics.If a system of N particles in 3d is subject to C holonomicconstraints, given by the equations f1, f2,..., fC, andthe dynamics given by a Lagrangian L(r, dr/dt, t), La-grange’s equations of the first kind are[11]
where k = 1, 2, ..., N labels the particles, there is a La-grange multiplier λi for each constraint equation fi, and
∂
∂rk≡(
∂
∂xk,
∂
∂yk,
∂
∂zk
),
∂
∂rk≡(
∂
∂xk,
∂
∂yk,
∂
∂zk
)
are each shorthands for a vector of derivatives with re-spect to the indicated variables (not a derivative with re-spect to the entire vector).[nb 1]
This procedure does increase the number of equations tosolve compared to Newton’s laws, from 3N to 3N + C,because there are 3N coupled second order differentialequations in the position coordinates and multipliers, plusC constraint equations. However, when solved alongsidethe position coordinates of the particles, the multiplierscan yield information about the constraint forces. Thecoordinates do not need to be eliminated by solving theconstraint equations.
12 CHAPTER 2. LAGRANGIAN MECHANICS
2.3.2 Lagrange’s equations of the secondkind
The Euler–Lagrange equations, or Lagrange’s equa-tions of the second kind[12][13]
are mathematical results from the calculus of variations,which can also be used in mechanics. Substituting in theLagrangian L(q, dq/dt, t), gives the equations of motionof the system.The number of equations has decreased compared toNewtonian mechanics, from 3N to n = 3N − C cou-pled second order differential equations in the general-ized coordinates. These equations do not include con-straint forces at all, only non-constraint forces need to beaccounted for.
2.4 From Newtonian to La-grangian mechanics
2.4.1 Newton’s laws
Isaac Newton (1642—1726)
For simplicity, Newton’s laws can be illustrated for oneparticle without much loss of generality (for a system ofN particles, all of these equations apply to each particlein the system). The equation of motion for particle of
massm is Newton’s second law of 1687, in modern vectornotation
F = ma ,
where a is its acceleration and F the resultant force act-ing on it. In three spatial dimensions, this is a systemof three coupled second order ordinary differential equa-tions to solve, since there are three components in thisvector equation. The solutions are the position vectors rof the particles at time t, subject to the initial conditionsof r and v when t = 0.Newton’s laws are easy to use in Cartesian coordinates,but Cartesian coordinates are not always convenient, andfor other coordinate systems the equations of motion canbecome complicated. In a set of curvilinear coordinatesξ = (ξ1, ξ2, ξ3), the law in tensor index notation is the“Lagrangian form” [14][15]
F a = m
(d2ξa
dt2+ Γa
bcdξb
dt
dξc
dt
)=
d
dt
∂T
∂ξa− ∂T
∂ξa, ξa ≡ dξa
dt,
where Fa is the ath contravariant components of the resul-tant force acting on the particle, Γabc are the Christoffelsymbols of the second kind,
T =1
2mgbc
dξb
dt
dξc
dt,
is the kinetic energy of the particle, and gbc the covariantcomponents of the metric tensor of the curvilinear coor-dinate system. All the indices a, b, c, each take the values1, 2, 3. Curvilinear coordinates are not the same as gen-eralized coordinates.It may seem like an overcomplication to cast Newton’slaw in this form, but there are advantages. The accel-eration components in terms of the Christoffel symbolscan be avoided by evaluating derivatives of the kineticenergy instead, a scalar invariant which takes the samevalue in all frames of reference. If there is no resultantforce acting on the particle, F = 0, it does not acceler-ate, but moves with constant velocity in a straight line.Mathematically, the solutions of the differential equationare geodesics, the curves of shortest length between twopoints in space. In flat 3d real space the geodesics aresimply straight lines. So for a free particle, Newton’s sec-ond law coincides with the geodesic equation, and statesfree particles follow geodesics, the shortest trajectories itcan move along. If the particle is subject to forces, F ≠0, the particle accelerates due to forces acting on it, anddeviates away from the geodesics it would follow if free.The idea of finding the shortest path a particle canfollow motivated the first applications of the calculusof variations to mechanical problems, such as theBrachistochrone problem solved by Jean Bernoulli in
2.4. FROM NEWTONIAN TO LAGRANGIAN MECHANICS 13
1696, as well as Leibniz, Daniel Bernoulli, L'Hôpitalaround the same time, and Newton the following year.[16]Newton himself was thinking along the lines of the varia-tional calculus, but did not publish.[17] These ideas in turnlead to the variational principles of mechanics, of Fermat,Maupertuis, Euler, Hamilton, and others. With appropri-ate extensions of the quantities given here in flat 3d spaceto 4d curved spacetime, the above form of Newton’s lawalso carries over to Einstein's general relativity, in whichcase free particles follow geodesics in curved spacetimethat are no longer “straight lines” in the ordinary sense.[18]
However, we still need to know the total resultant force Facting on the particle, which in turn requires the resultantnon-constraint force N plus the resultant constraint forceC,
F = C+ N .
The constraint forces can be complicated, since they willgenerally depend on time. Also, if there are constraints,the curvilinear coordinates are not independent but re-lated by one or more constraint equations.The constraint forces can either be eliminated from theequations of motion so only the non-constraint forces re-main, or included by including the constraint equations inthe equations of motion.
2.4.2 D'Alembert’s principle
Jean d'Alembert (1717—1783)
m
C
N
q
f(x,y)=0
One degree of freedom.
N
C
q 1
q 2
f(x,y,z)=0
m
q 1
q 2
Two degrees of freedom.Constraint force C and virtual displacement δr for aparticle of mass m confined to a curve. The resultantnon-constraint force is N.
A fundamental result in analytical mechanics isD'Alembert’s principle, introduced in 1708 by JacquesBernoulli to understand static equilibrium, and developedby D'Alembert in 1743 to solve dynamical problems.[19]The principle asserts for N particles[10]
N∑k=1
(Nk + Ck −mkak) · δrk = 0 .
The δrk are virtual displacements, by definition they areinfinitesimal changes in the configuration of the systemconsistent with the constraint forces acting on the sys-tem at an instant of time,[20] i.e. in such a way that theconstraint forces maintain the constrained motion. Theyare not the same as the actual displacements in the sys-tem, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate andmove it.[nb 2] Virtual work is the work done along a virtualdisplacement for any force (constraint or non-constraint).Since the constraint forces act perpendicular to the mo-tion of each particle in the system to maintain the con-straints, the total virtual work by the constraint forces act-ing on the system is zero;[21]
N∑k=1
Ck · δrk = 0 ,
so that
14 CHAPTER 2. LAGRANGIAN MECHANICS
N∑k=1
(Nk −mkak) · δrk = 0 .
Thus D'Alembert’s principle allows us to concentrate ononly the applied non-constraint forces, and exclude theconstraint forces in the equations of motion.[22][23] Theform shown is also independent of the choice of coor-dinates. However, it is not readily usable to set up theequations of motion and solve for the motion of the sys-tem; therefore equations ofmotion in a set of independentcoordinates are sought for.
2.4.3 Equations of motion fromD'Alembert’s principle
If there are constraints on particle k, then since the coor-dinates of the position rk = (xk, yk, zk) are linked togetherby a constraint equation, so are the those of the virtual dis-placements δrk = (δxk, δyk, δzk). Since the generalizedcoordinates are independent, we can avoid the complica-tions with the δrk by converting to virtual displacementsin the generalized coordinates. These are related in thesame form as a total differential,[24]
δrk =n∑
j=1
∂rk∂qj
δqj .
There is no partial time derivative with respect to timemultiplied by a time increment, since this is a virtual dis-placement, one along the constraints in an instant of time.The first term in D'Alembert’s principle above is the vir-tual work done by the non-constraint forces Nk along thevirtual displacements δrk, and can without loss of gener-ality be converted into the generalized analogues by thedefinition of generalized forces
Qj =N∑
k=1
Nk · ∂rk∂qj
,
so that
N∑k=1
Nk · δrk =N∑
k=1
Nk ·n∑
j=1
∂rk∂qj
δqj =n∑
j=1
Qjδqj .
This is half of the conversion to generalized coordinates.It remains to convert the acceleration term into general-ized coordinates, which is not immediately obvious. Re-calling the Lagrange form of Newton’s second law, thepartial derivatives of the kinetic energy with respect tothe generalized coordinates and velocities can be foundto give the desired result;[25]
N∑k=1
mkak · ∂rk∂qj
=d
dt
∂T
∂qj− ∂T
∂qj.
Now D'Alembert’s principle is in the generalized coordi-nates as required,
n∑j=1
[Qj −
(d
dt
∂T
∂qj− ∂T
∂qj
)]δqj = 0 ,
and since these virtual displacements δqj are independentand nonzero, the coefficients can be equated to zero, re-sulting in Lagrange’s equations[26][27] or the general-ized equations of motion,[28]
Qj =d
dt
∂T
∂qj− ∂T
∂qj
These equations are equivalent to Newton’s laws for thenon-constraint forces. The generalized forces in thisequation are derived from the non-constraint forces only- the constraint forces they have been excluded fromD'Alembert’s principle and do not need to be found. Thegeneralized forces may be non-conservative, providedthey satisfy D'Alembert’s principle.[29]
2.4.4 Euler–Lagrange equations andHamilton’s principle
As the system evolves, q traces a path through configuration space(only some are shown). The path taken by the system (red) hasa stationary action (δS = 0) under small changes in the configu-ration of the system (δq).[30]
For a non-conservative force which depends on velocity,it may be possible to find a potential energy function Vthat depends on positions and velocities. If the gener-alized forces Qi can be derived from a potential V suchthat[31][32]
Qj =d
dt
∂V
∂qj− ∂V
∂qj,
2.4. FROM NEWTONIAN TO LAGRANGIAN MECHANICS 15
equating to Lagrange’s equations and defining the La-grangian as L = T − V obtains Lagrange’s equationsof the second kind or the Euler–Lagrange equationsof motion
∂L
∂qj− d
dt
∂L
∂qj= 0 .
However, the Euler–Lagrange equations can only accountfor non-conservative forces if a potential can be found asshown. Lagrange’s equations do not involve any poten-tial, only forces; therefore they are more general than theEuler–Lagrange equations.These equations also follow from the calculus of varia-tions. The variation of the Lagrangian is
δL =n∑
j=1
(∂L
∂qjδqj +
∂L
∂qjδqj
), δqj ≡ δ
dqjdt
≡ d(δqj)
dt,
which has a similar form to the total differential of L, butthe virtual displacements and their time derivatives re-place differentials, and there is no time increment in ac-cordance with the definition of the virtual displacements.An integration by parts with respect to time can trans-fer the time derivative of δqj to the ∂L/∂(dqj/dt), in theprocess exchanging d(δqj)/dt for δqj, allowing the inde-pendent virtual displacements to be factorized from thederivatives of the Lagrangian,
∫ t2
t1
δL dt =n∑
j=1
[∂L
∂qjδqj
]t2t1
+
∫ t2
t1
n∑j=1
(∂L
∂qj− d
dt
∂L
∂qj
)δqj dt .
Now, if the condition δqj(t1) = δqj(t2) = 0 holds for all j,the terms not integrated are zero. If in addition the entiretime integral of δL is zero, then because the δqj are inde-pendent, and the only way for a definite integral to be zerois if the integrand equals zero, each of the coefficients ofδqj must also be zero. Then we obtain the equations ofmotion. This can be summarized by Hamilton’s princi-ple;
∫ t2
t1
δL dt = 0 .
The time integral of the Lagrangian is another quantitycalled the action, defined as[33]
S =
∫ t2
t1
Ldt ,
which is a functional; it takes in the Lagrangian functionfor all times between t1 and t2 and returns a scalar value.Its dimensions are the same as [ angular momentum ],
[energy]·[time], or [length]·[momentum]. With this def-inition Hamilton’s principle is
δS = 0 .
Thus, instead of thinking about particles accelerating inresponse to applied forces, one might think of them pick-ing out the path with a stationary action, with the endpoints of the path in configuration space held fixed at theinitial and final times. Hamilton’s principle is sometimesreferred to as the principle of least action, however theaction functional need only be stationary, not necessar-ily a maximum or a minimum value. Any variation ofthe functional gives an increase in the functional inte-gral of the action. It is not widely stated that Hamilton’sprinciple is a variational principle only with holonomicconstraints, if we are dealing with nonholonomic systemsthen the variational principle should be replaced with oneinvolving d'Alembert principle of virtual work.
2.4.5 Lagrange multipliers and con-straints
We can vary L in the Cartesian rk coordinates, for N par-ticles,
∫ t2
t1
N∑k=1
(∂L
∂rk− d
dt
∂L
∂rk
)· δrk dt = 0 .
Hamilton’s principle is still valid even if the coordinatesL is expressed in are not independent, here rk, but theconstraints are still assumed to be holonomic.[34] Whatcannot be done is to simply equate the coefficients of δrkto zero because the δrk are not independent. Instead, themethod of Lagrangemultipliers can be used to include theconstraints. Multiplying each constraint equation fi(rk, t)= 0 by a Lagrange multiplier λi for i = 1, 2, ..., C, andadding the results to the original Lagrangian, gives thenew Lagrangian
L′ = L(r1, r2, . . . , r1, r2, . . . , t) +C∑i=1
λi(t)fi(rk, t) .
The Lagrange multipliers are arbitrary functions of timet, but not functions of the coordinates rk, so the multi-pliers are on equal footing with the position coordinates.With this new Lagrangian the Euler-Lagrange equationsrecover the constraint equations
∂L′
∂λi− d
dt
∂L′
∂λi
= 0 ⇒ fi(rk, t) = 0
and also obtain the equations of motion in terms of therk, which are Lagrange’s equations of the first kind
16 CHAPTER 2. LAGRANGIAN MECHANICS
∂L′
∂rk− d
dt
∂L′
∂rk= 0 ⇒ ∂L
∂rk− d
dt
∂L
∂rk+
C∑i=1
λi∂fi∂rk
= 0 ,
These equations also follow by varying the new La-grangian and integrating over time;
∫ t2
t1
N∑k=1
(∂L
∂rk− d
dt
∂L
∂rk+
C∑i=1
λi∂fi∂rk
)· δrk dt = 0 .
where as always the end points are fixed, δrk(t1) = δrk(t2)= 0 for all k. The multipliers can be found so that thecoefficients of δrk are zero, even though the rk are notindependent. The equations of motion follow.For the case of a conservative force given by the gradientof some potential energy V, a function of the r coordi-nates only, substituting the Lagrangian L = T − V gives
∂T
∂rk− d
dt
∂T
∂rk− ∂V
∂rk+
C∑i=1
λi∂fi∂rk
= 0 ,
From the Lagrangian form ofNewton’s second law above,the derivatives of kinetic energy form the (negative of)resultant force acting on the particle k, and the derivativesof the potential form the non-constraint force acting onthe particle k,
−Fk =∂T
∂rk− d
dt
∂T
∂rk, Nk = − ∂V
∂rk,
so it follows the constraint forces are given by
Ck =
C∑i=1
λi∂fi∂rk
,
which relates the constraint equations to the constraintforces via the Lagrange multipliers.
2.5 Properties of the Euler–Lagrange equation
In some cases, the Lagrangian has properties which canprovide information about the system without solving theequations of motion. These follow from Lagrange’s equa-tions of the second kind.
2.5.1 Non uniqueness
The Lagrangian of a given system is not unique. A La-grangian L can be multiplied by a nonzero constant a,
an arbitrary constant b can be added, and the new La-grangian aL + b will describe exactly the same motion asL. A less obvious result is that two Lagrangians describ-ing the same system can differ by the total derivative (notpartial) of some function f(q, t) with respect to time;[35]
L′ = L+df(q, t)
dt.
Each Lagrangian will obtain exactly the same equationsof motion.[36][37]
2.5.2 Invariance under point transforma-tions
Given a set of generalized coordinates q, if we changethese variables to a new set of generalized coordinates saccording to a point transformation q = q(s, t), the newLagrangian L′ is a function of the new coordinates
L(q(s, t), q(s, s, t), t) = L′(s, s, t) ,
and by the chain rule for partial differentiation,Lagrange’s equations are invariant under thistransformation;[38]
d
dt
∂L′
∂si=
∂L′
∂si.
This may simplify the equations of motion. The pro-cedure is analogous to Canonical transformations inHamiltonian mechanics, which preserves the form ofHamiltonian equations.
2.5.3 Cyclic coordinates and conservationlaws
An important property of the Lagrangian is thatconserved quantities can easily be read off from it. Thegeneralized momentum “canonically conjugate to” the co-ordinate qi is defined by
pi =∂L
∂qi.
If the Lagrangian L does not depend on some coordinateqi, then it follows from the Euler–Lagrange equations thatthe corresponding generalized momentum will be a con-served quantity, because its time derivative is zero so themomentum must be a constant of the motion;
pi =d
dt
∂L
∂qi=
∂L
∂qi= 0 .
2.5. PROPERTIES OF THE EULER–LAGRANGE EQUATION 17
This is a special case of Noether’s theorem. Such coor-dinates are called “cyclic” or “ignorable”.For example, a system may have a Lagrangian
L(r, θ, s, z, r, θ, ϕ, t) ,
where r and z are lengths along straight lines, s is an arclength along some curve, and θ and φ are angles. Noticez, s, and φ are all absent in the Lagrangian even thoughtheir velocities are not. Then the momenta
pz =∂L
∂z, ps =
∂L
∂s, pϕ =
∂L
∂ϕ,
are all conserved quantities. The units and nature of eachgeneralized momentumwill depend on the correspondingcoordinate; in this case pz is a translational momentum inthe z direction, ps is also a translational momentum alongthe curve s is measured, and pφ is an angular momen-tum in the plane the angle φ is measured in. Whateverthe complicated the motion of the system is; all the co-ordinates and velocities will vary in such a way that thesemomenta are conserved.
2.5.4 Energy conservation
Taking the total derivative of the Lagrangian
L = T − V
with respect to time leads to the general result
∂L
∂t=
d
dt
(n∑
i=1
qi∂L
∂qi− L
).
If the entire Lagrangian is explicitly independent of time,it follows the partial time derivative of the Lagrangianis zero, ∂L/∂t = 0, so the quantity under the total timederivative in brackets
E =n∑
i=1
qi∂L
∂qi− L
must be a constant for all times during the motion ofthe system, and it also follows the kinetic energy is ahomogenous function of degree 2 in the generalized ve-locities. If in addition the potential V is only a functionof coordinates and independent of velocities, it follows bydirect calculation, or use of Euler’s theorem for homoge-nous functions, that
n∑i=1
qi∂L
∂qi=
n∑i=1
qi∂T
∂qi= 2T .
Under all these circumstances,[39] the constant
E = 2T − L = T + V
is the total conserved energy of the system. The kineticand potential energies still change as the system evolves,but the motion of the system will be such that their sum,the total energy, is constant. This is a valuable simpli-fication, since the energy E is a constant of integrationthat counts as an arbitrary constant for the problem, andit may be possible to integrate the velocities from this en-ergy relation to solve for the coordinates. In the case thevelocity or kinetic energy or both depends on time, thenthe energy is not conserved.The Hamiltonian is related to the Lagrangian by aLegendre transformation. By definition it is the abovequantity in brackets,
H =n∑
i=1
qi∂L
∂qi− L .
Under the same conditions, the Hamiltonian equals thetotal energy of the system and is conserved.
2.5.5 Mechanical similarity
Main article: Mechanical similarity
If the potential energy is a homogeneous function of thecoordinates and independent of time,[40] and all positionvectors are scaled by the same nonzero constant α, rk′ =αrk, so that
V (αr1, αr2, . . . , αrN ) = αNV (r1, r2, . . . , rN )
and time is scaled by a factor β, t′ = βt, then the velocitiesvk are scaled by a factor of α/β and the kinetic energy Tby (α/β)2. The entire Lagrangian has been scaled by thesame factor if
α2
β2= αN ⇒ β = α1−N/2 .
Since the lengths and times have been scaled, the trajec-tories of the particles in the system follow geometricallysimilar paths differing in size. The length l traversedin time t in the original trajectory corresponds to a newlength l′ traversed in time t′ in the new trajectory, givenby the ratios
t′
t=
(l′
l
)1−N/2
.
18 CHAPTER 2. LAGRANGIAN MECHANICS
2.5.6 Interacting particles
For a given system, if two subsystems A and B are non-interacting, the Lagrangian L of the overall system is thesum of the Lagrangians LA and LB for the subsystems:[35]
L = LA + LB .
If they do interact this is not possible. In some situa-tions, it may be possible to separate the Lagrangian ofthe system L into the above form of non-interacting La-grangians, plus another Lagrangian LAB containing infor-mation about their intertwined motion and the potentialenergy for the interaction,
L = LA + LB + LAB .
This can be physically motivated from the limiting case ofnegligible interaction - then interaction Lagrangian tendsto zero reducing to the non-interacting case above.The extension to more than two non-interacting subsys-tems is straightforwards - the overall Lagrangian is thesum of the separate Lagrangians for each subsystem. Ifthere are interactions, then interaction Lagranians may beadded.
2.6 Examples in specific coordinatesystems
In the following examples, a particle of mass m movesunder the influence of a conservative force derived fromthe gradient ∇ of the a scalar potential,
F = −∇V (r) .
If there are more particles, in accordance with the aboveresults, the total kinetic energy is a sum over all the parti-cle kinetic energies, and the potential is a function of allthe coordinates.
2.6.1 Cartesian coordinates
The Lagrangian of the particle can be written
L(x, y, z, x, y, z) =1
2m(x2 + y2 + z2)− V (x, y, z) .
The equations of motion for the particle are found by ap-plying the Euler–Lagrange equation, for the x coordinate
d
dt
(∂L
∂x
)− ∂L
∂x= 0,
and similarly for the y and z coordinates. For the x coor-dinate
∂L
∂x= −∂V
∂x,
∂L
∂x= mx ,
d
dt
(∂L
∂x
)= mx ,
hence
mx = −∂V
∂x.
and similarly for the y and z coordinates. Collecting theequations in vector form we find
mr = −∇V
which is Newton’s second law of motion for a particlesubject to a conservative force.
2.6.2 Polar coordinates in 2d and 3d
The Lagrangian for the above problem in spherical coor-dinates is
L =m
2(r2 + r2θ2 + r2 sin2 θ φ2)− V (r) ,
so the Euler–Lagrange equations are
mr −mr(θ2 + sin2 θ φ2) +∂V
∂r= 0 ,
d
dt(mr2θ)−mr2 sin θ cos θ φ2 = 0 ,
d
dt(mr2 sin2 θ φ) = 0 .
The φ coordinate is cyclic since it does not appear in theLagrangian, so the conserved momentum in the system isthe angular momentum
pφ =∂L
∂φ= mr2 sin2 θφ ,
in which r, θ and dφ/dt can all vary with time, but only insuch a way that pφ is constant.
2.7 Non-relativistic examples
The following examples apply Lagrange’s equations of thesecond kind to mechanical problems.
2.7. NON-RELATIVISTIC EXAMPLES 19
2.7.1 Pendulum on a movable support
Consider a pendulum of mass m and length ℓ, which isattached to a support with massM, which can move alonga line in the x-direction. Let x be the coordinate along theline of the support, and let us denote the position of thependulum by the angle θ from the vertical.
Sketch of the situation with definition of the coordinates (click toenlarge)
The kinetic energy can then be shown to be
T = 12Mx2 + 1
2m(x2pend + y2pend
)= 1
2Mx2 + 12m
[(x+ ℓθ cos θ
)2+(ℓθ sin θ
)2],
and the potential energy of the system is
V = mgypend = −mgℓ cos θ .
The Lagrangian is therefore
L = T − V
= 12Mx2 + 1
2m
[(x+ ℓθ cos θ
)2+(ℓθ sin θ
)2]+mgℓ cos θ
= 12 (M +m) x2 +mxℓθ cos θ + 1
2mℓ2θ2 +mgℓ cos θ
Since x is absent from the Lagrangian, it is a cyclic coor-dinate. The conserved momentum is
px = (M +m)x+mℓθ cos θ .
The Lagrange equation for the support coordinate x istherefore
d
dt
[(M +m)x+mℓθ cos θ
]= 0,
or
(M +m)x+mℓθ cos θ −mℓθ2 sin θ = 0
The Lagrange equation for the angle θ is
d
dt
[m(xℓ cos θ + ℓ2θ)
]+mℓ(xθ + g) sin θ = 0;
therefore
θ +x
ℓcos θ + g
ℓsin θ = 0.
These equations may look quite complicated, but findingthem with Newton’s laws would have required carefullyidentifying all forces, which would have been much morelaborious and prone to errors. By considering limit cases,the correctness of this system can be verified: For ex-ample, x → 0 should give the equations of motion fora pendulum that is at rest in some inertial frame, whileθ → 0 should give the equations for a pendulum in a con-stantly accelerating system, etc. Furthermore, it is trivialto obtain the results numerically, given suitable startingconditions and a chosen time step, by stepping throughthe results iteratively.
2.7.2 Two-body central force problem
Main articles: Two-body problem and Central force
The basic problem is that of two bodies of massesm1 andm2 with position vectors r1 and r2 are in orbit about eachother due to an attractive central force V. We may naïvelywrite down the Lagrangian in terms of the position coor-dinates as they are, but it is an established procedure toconvert the two-body problem into a one-body problemas follows. Introduce the Jacobi coordinates; the separa-tion of the bodies r = r2 − r1 and the location of the centerof mass R = (m1r1 + m2r2)/(m1 + m2). The Lagrangianis then[41][42][nb 3]
L = T − U =1
2M R2 +
(1
2µr2 − V (|r|)
)= Lcm + Lrel
where M = m1 + m2 is the total mass, μ = m1m2/(m1
+ m2) is the reduced mass, and V the potential of theradial force, which depends only on the magnitude of theseparation |r| = |r2 − r1|. The Lagrangian is divided intoa center-of-mass term L and a relative motion term Lᵣₑ .The Euler–Lagrange equation for R is simply
M R = 0 ,
20 CHAPTER 2. LAGRANGIAN MECHANICS
which states the center of mass moves in a straight lineat constant velocity. The since the relative motion onlydepends on the magnitude of the separation, it is ideal touse polar coordinates (r, θ) and take r = |r|,
L =1
2µ(r2 + r2θ2)− V (r) ,
which does not depend upon θ, therefore θ is an ignorablecoordinate. The conserved momentum corresponding toθ is
pθ =∂L
∂θ= µr2θ = ℓ ,
which will be abbreviated ℓ. The radial coordinate r andangular velocity dθ/dt can vary with time, but only in sucha way that ℓ is constant. The Lagrange equation for r is
∂L
∂r=
d
dt
∂L
∂r⇒ µrθ2 − dU
dr= µr .
This equation is identical to the radial equation obtainedusing Newton’s laws in a co-rotating reference frame, thatis, a frame rotating with the reduced mass so it appearsstationary. Eliminating the angular velocity dθ/dt fromthis radial equation,[43]
µr = −dU
dr+
ℓ2
µr3.
which is the equation of motion for a one-dimensionalproblem in which a particle of mass μ is subjected to theinward central force −dU/dr and a second outward force,called in this context the centrifugal force
Fcf = µrθ2 =ℓ2
µr3.
Of course, if one remains entirely within the one-dimensional formulation, ℓ enters only as some imposedparameter of the external outward force, and its inter-pretation as angular momentum depends upon the moregeneral two-dimensional problem from which the one-dimensional problem originated.If one arrives at this equation using Newtonian mechan-ics in a co-rotating frame, the interpretation is evident asthe centrifugal force in that frame due to the rotation ofthe frame itself. If one arrives at this equation directlyby using the generalized coordinates (r, θ) and simplyfollowing the Lagrangian formulation without thinkingabout frames at all, the interpretation is that the centrifu-gal force is an outgrowth of using polar coordinates. AsHildebrand says:[44]
“Since such quantities are not true physical forces, theyare often called inertia forces. Their presence or ab-sence depends, not upon the particular problem at hand,but upon the coordinate system chosen.” In particular, ifCartesian coordinates are chosen, the centrifugal forcedisappears, and the formulation involves only the cen-tral force itself, which provides the centripetal force for acurved motion.This viewpoint, that fictitious forces originate in thechoice of coordinates, often is expressed by users of theLagrangian method. This view arises naturally in theLagrangian approach, because the frame of reference is(possibly unconsciously) selected by the choice of coor-dinates. For example, see[45] for a comparison of La-grangians in an inertial and in a noninertial frame of ref-erence. See also the discussion of “total” and “updated”Lagrangian formulations in.[46] Unfortunately, this usageof “inertial force” conflicts with the Newtonian idea of aninertial force. In the Newtonian view, an inertial forceoriginates in the acceleration of the frame of observation(the fact that it is not an inertial frame of reference), not inthe choice of coordinate system. To keep matters clear, itis safest to refer to the Lagrangian inertial forces as gener-alized inertial forces, to distinguish them from the New-tonian vector inertial forces. That is, one should avoidfollowing Hildebrand when he says (p. 155) “we deal al-wayswith generalized forces, velocities accelerations, andmomenta. For brevity, the adjective “generalized” will beomitted frequently.”It is known that the Lagrangian of a system is not unique.Within the Lagrangian formalism the Newtonian ficti-tious forces can be identified by the existence of alter-native Lagrangians in which the fictitious forces disap-pear, sometimes found by exploiting the symmetry of thesystem.[47]
2.7.3 Non-relativistic test particles in fields
A test particle is a particle whose mass and charge areassumed to be so small that its effect on external systemis insignificant. It is often a hypothetical simplified pointparticle with no properties other than mass and charge.Real particles like electrons and up quarks are more com-plex and have additional terms in their Lagrangians.
Non-relativistic test particle in a Newtonian gravita-tional field
For a particle with massm in a Newtonian gravitation po-tential
V (r(t), t) = mΦ(r(t), t) ,
since the force is conservative, one can follow the sameprocedure in the Cartesian coordinates example to find
2.9. DEFINITIONS OF THE LAGRANGIAN (RELATIVISTIC MECHANICS) 21
r = −∇Φ
where the mass cancels algebraically, and physically itshould because the acceleration of all massive objects dueto gravity is independent of the mass.
Non-relativistic test particle in an electromagneticfield
The case of a charged particle with electrical charge qinteracting with an electromagnetic field is more compli-cated. The electric scalar potentialϕ(r(t), t) andmagneticvector potential A(r(t), t) are defined from the electricfield
E(r(t), t) = −∇ϕ(r(t), t)− ∂A(r(t), t)∂t
,
and magnetic field
B(r(t), t) = ∇× A(r(t), t) .
Notice all the fields depend on the position r of the par-ticle at time t, and depend explicitly on time as well asimplicitly via the position.The Lagrangian of a massive charged test particle in anelectromagnetic field is
L =m
2r2 − qϕ+ qr · A ,
which produces the Lorentz force law
mr = qE+ qr× B .
An interesting point in this example is the generalizedmomentum conjugate to r is the ordinary momentumplus a contribution from the A field,
p =∂L
∂r = mr+ qA .
This relation is used in the minimal coupling prescriptionin quantum mechanics and quantum field theory.
2.8 Extensions to include non-conservative forces
Dissipation (i.e. non-conservative systems) can also betreated with an effective Lagrangian formulated by a cer-tain doubling of the degrees of freedom; see.[48][49][50][51]
In a more general formulation, the forces could be bothconservative and viscous. If an appropriate transforma-tion can be found from the Fᵢ, Rayleigh suggests using adissipation function, D, of the following form:[52]
D =1
2
m∑j=1
m∑k=1
Cjkqj qk .
where Cjk are constants that are related to the dampingcoefficients in the physical system, though not necessarilyequal to them. If D is defined this way, then[52]
Qj = −∂V
∂qj− ∂D
∂qj
and
d
dt
(∂L
∂qj
)− ∂L
∂qj+
∂D
∂qj= 0 .
2.9 Definitions of the Lagrangian(relativistic mechanics)
Lagrangian mechanics can be formulated in special rel-ativity as follows. Consider one particle (N particles areconsidered later).
2.9.1 Coordinate formulation
The Euler–Lagrange equations retain their form in specialrelativity, provided the Lagrangian generates equations ofmotion consistent with special relativity. It is possible totransform to generalized coordinates exactly as in non-relativistic mechanics, r = r(q, t). The velocity v in termsof the generalized coordinates and velocities remains thesame. However, the energy of a moving particle is dif-ferent to non-relativistic mechanics. It is instructive tolook at the total relativistic energy of a free test particle.Expanding in a power series, the first term is the parti-cle’s rest energy, plus its non-relativistic kinetic energy,followed by higher order relativistic corrections;
E = m0c2 dt
dτ=
m0c2√
1− r2(t)c2
= m0c2+
1
2m0r2(t)+
3
8m0
r4(t)c2
+. . . .
where c is the speed of light in vacuum, v = dr/dt is thecoordinate velocity of the particle as measured in somelab frame, t is the coordinate time in the lab frame, and τis the proper time (the time measured by a clock movingwith the particle). The differentials in t and τ are relatedby the Lorentz factor γ,[nb 4]
22 CHAPTER 2. LAGRANGIAN MECHANICS
dt = γ(r)dτ , γ(r) = 1√1− r2
c2
, r = drdt
, r2(t) = r(t)·r(t) .
(The overdots are with respect to coordinate time, notproper time). The relativistic kinetic energy for an un-charged particle of rest mass m0 is
T = (γ(r)− 1)m0c2
and we may naïvely guess the relativistic Lagrangian fora particle to be this relativistic kinetic energy minus thepotential energy. However, even for a free particle forwhich V = 0, this is wrong. Following the non-relativisticapproach, we expect the derivative of this seemingly cor-rect Lagrangian with respect to the velocity to be the rel-ativistic momentum, which it is not.The definition of a generalized momentum can be re-tained, and the advantageous connection between cycliccoordinates and conserved quantities will continue to ap-ply. The momenta can be used to “reverse-engineer” theLagrangian. For the case of the free massive particle,in Cartesian coordinates, the x component of relativisticmomentum is
px =∂L
∂x= γ(r)m0x ,
and similarly for the y and z components. Integrating thisequation with respect to dx/dt gives
L = −m0c2
γ(r) +X(y, z) ,
where X is an arbitrary function of dy/dt and dz/dt fromthe integration. Integrating py and pz obtains similarly
L = −m0c2
γ(r) + Y (x, z) , L = −m0c2
γ(r) + Z(x, y) ,
where Y and Z are arbitrary functions of their indicatedvariables. Since the functions X, Y, Z are arbitrary, with-out loss of generality we can conclude the common so-lution to these integrals, a possible Lagrangian that willcorrectly generate all the components of relativistic mo-mentum, is
L = −m0c2
γ(r) ,
where X = Y = Z = 0.Alternatively, since we wish to build a Lagrangian out ofrelativistically invariant quantities, take the action as pro-portional to the integral of the Lorentz invariant line ele-ment in spacetime, the length of the particle’s world linebetween proper times τ1 and τ2,[nb 4]
S = ε
∫ τ2
τ1
dτ = ε
∫ t2
t1
dt
γ(r) , L =ε
γ(r) = ε
√1− r2
c2,
where ε is a constant to be found, and after converting theproper time of the particle to the coordinate time as mea-sured in the lab frame, the integrand is the Lagrangian bydefinition. The momentum must be the relativistic mo-mentum,
p =∂L
∂r =
(−ε
c2
)γ(r)r = m0γ(r)r ,
which requires ε = −m0c2, in agreement with the previ-ously obtained Lagrangian.Either way, the position vector r is absent from the La-grangian and therefore cyclic, so the Euler–Lagrangeequations are consistent with the constancy of relativis-tic momentum,
d
dt
∂L
∂r =∂L
∂r ⇒ d
dt(m0γ(r)r) = 0 ,
which must be the case for a free particle. Also, expand-ing the relativistic free particle Lagrangian to first orderin (v/c)2,
L = −m0c2
[1 +
1
2
(− r2c2
)+ · · ·
]≈ −m0c
2+m0
2r2 ,
in the non-relativistic limit when v is small, the higher or-der terms not shown are negligible, and the Lagrangian isthe non-relativistic kinetic energy as it should be. The re-maining term is the negative of the particle’s rest energy,a constant term which can be ignored in the Lagrangian.For the case of an interacting particle subject to a poten-tial V, which may be non-conservative, it is possible fora number of interesting cases to simply subtract this po-tential from the free particle Lagrangian,
L = −m0c2
γ(r) − V (r, r, t) .
and the Euler–Lagrange equations lead to the relativis-tic version of Newton’s second law, the coordinate timederivative of relativistic momentum is the force acting onthe particle;
F =d
dt
∂V
∂r − ∂V
∂r =d
dt(m0γ(r)r) .
assuming the potential V can generate the correspondingforce F in this way.
2.9. DEFINITIONS OF THE LAGRANGIAN (RELATIVISTIC MECHANICS) 23
It is also true that if the Lagrangian is explicitly indepen-dent of time and the potential V(r) independent of veloc-ities, then the total relativistic energy
E =∂L
∂r · r− L = γ(r)m0c2 + V (r)
is conserved, although the identification is less obvioussince the first term is the relativistic energy of the particlewhich includes the rest mass of the particle, not merelythe relativistic kinetic energy. Also, the argument forhomogenous functions does not apply to relativistic La-grangians.The extension to N particles is streightforward, the rel-ativistic Lagrangian is just a sum of the “free particle”terms, minus the potential energy of their interaction;
L = −c2N∑
k=1
m0k
γ(rk)− V (r1, r2, . . . , r1, r2, . . . , t) ,
where all the positions and velocities are measured in thesame lab frame, including the time.The advantage of this coordinate formulation is that it canbe applied to a variety of systems, including multiparti-cle systems. The disadvantage is that some lab frame hasbeen singled out as a preferred frame, and none of theequations are manifestly covariant (in other words, theydo not take the same form in all frames of reference).For an observer moving relative to the lab frame, every-thing must be recalculated using Lorentz transformationsof all the quantities; the position r, the momentum p, to-tal energy E, potential energy, etc. The action will remainthe same since it is Lorentz invariant by construction.A seemingly different but completely equivalent form ofthe Lagrangian for a free massive particle, which willreadily extend to general relativity as shown below, canbe obtained by inserting[nb 4]
dτ =1
c
√ηαβ
dxα
dt
dxβ
dtdt ,
into the Lorentz invariant action so that
S = ε
∫ t2
t1
1
c
√ηαβ
dxα
dt
dxβ
dtdt ⇒ L =
ε
c
√ηαβ
dxα
dt
dxβ
dt
where ε = −m0c2 is retained for simplicity. Althoughthe line element and action are Lorentz invariant, the La-grangian is not, because it has explicit dependence on thelab coordinate time. Still, this route illustrates that find-ing the stationary action is akin to finding the trajectoryof shortest or largest length in spacetime, and correspond-ingly the equations of motion of the particle are akin to
the equations describing the trajectories of shortest orlargest length in spacetime, i.e. geodesics.For the case of an interacting particle in a potential V, theLagrangian is still
L =ε
c
√ηαβ
dxα
dt
dxβ
dt− V .
2.9.2 Covariant formulation
In the covariant formulation, time is placed on equalfooting with space, so the coordinate time as mea-sured in some frame is part of the configuration spacealongside the spatial coordinates (and other generalizedcoordinates).[53] For a particle, either massless or mas-sive, the Lorentz invariant action is (abusing notation)[54]
S =
∫ σ2
σ1
Λ(xν(σ), uν(σ), σ)dσ
where lower and upper indices are used according tocovariance and contravariance of vectors, σ is an affineparameter, and uμ = dxμ/dσ is the four-velocity of the par-ticle.For massive particles, σ can be the arc length s, or propertime τ, along the particle’s world line,
ds2 = c2dτ2 = gαβdxαdxβ .
For massless particles, it cannot because the proper timeof a massless particle is always zero;
gαβdxαdxβ = 0 .
For a free particle, the Lagrangian has the form[55][56]
Λ = gαβdxα
dσ
dxβ
dσ
where the irrelevant factor of 1/2 is allowed to be scaledaway by the scaling property of Lagrangians. No in-clusion of mass is necessary since this also applies tomassless particles. The Euler–Lagrange equations in thespacetime coordinates are
d
dσ
∂Λ
∂uα− ∂Λ
∂xα=
d2xα
dσ2+ Γα
βγdxβ
dσ
dxγ
dσ= 0 ,
which is the geodesic equation for affinely parameterizedgeodesics in spacetime. In other words, the free particlefollows geodesics. Geodesics for massless particles arecalled “null geodesics”, since they lie in a "light cone" or
24 CHAPTER 2. LAGRANGIAN MECHANICS
“null cone” of spacetime, while those for massive parti-cles are called “non-null geodesics”.This manifestly covariant formulation does not extend toan N particle system, since then the affine parameter ofany one particle cannot be defined as a common param-eter for all the other particles.
2.10 Relativistic examples
2.10.1 Special relativistic 1d harmonic os-cillator
For a 1d relativistic simple harmonic oscillator, the La-grangian is[57][58]
L = −mc2√1− x2(t)
c2− k
2x2 .
where k is the spring constant.
2.10.2 Special relativistic constant force
For a particle under a constant force, the Lagrangian is[59]
L = −mc2√1− x2(t)
c2−max .
where a is the force per unit mass.
2.10.3 Special relativistic test particle in anelectromagnetic field
Main article: Covariant formulation of classical electro-magnetism
In special relativity, the Lagrangian of a massive chargedtest particle in an electromagnetic field modifies to[60]
L = −mc2√1− v2
c2− qϕ+ qr · A .
The Lagrangian equations in r lead to the Lorentz forcelaw, in terms of the relativistic momentum
d
dt
mr√1− v2
c2
= qE+ qr× B .
In the language of four vectors and tensor index notation,the Lagrangian takes the form
L(τ) =1
2muµ(τ)uµ(τ) + quµ(τ)Aµ(x)
where uμ = dxμ/dτ is the four-velocity of the test particle,and Aμ the electromagnetic four potential.The Euler–Lagrange equations are (notice the to-tal derivative with respect to proper time instead ofcoordinate time)
∂L
∂xν− d
dτ
∂L
∂uν= 0
obtains
quµ ∂Aµ
∂xν=
d
dτ(muν + qAν) .
Under the total derivative with respect to proper time, thefirst term is the relativistic momentum, the second termis
dAν
dτ=
∂Aν
∂xµ
dxµ
dτ=
∂Aν
∂xµuµ ,
then rearranging, and using the definition of the antisym-metric electromagnetic tensor, gives the covariant formof the Lorentz force law in the more familiar form,
d
dτ(muν) = quµF ν
µ , F νµ =
∂Aµ
∂xν− ∂Aν
∂xµ.
2.10.4 General relativistic test particle inan electromagnetic field
In general relativity, the first term generalizes (includes)both the classical kinetic energy and the interaction withthe gravitational field. For a charged particle in an elec-tromagnetic field it is
L(t) = −mc2√−c−2gµν(x(t))
dxµ(t)
dt
dxν(t)
dt+q
dxµ(t)
dtAµ(x(t)) .
If the four spacetime coordinates xµ are given in arbitraryunits (i.e. unitless), then gµν in m2 is the rank 2 symmet-ric metric tensor which is also the gravitational potential.Also, Aµ in V·s is the electromagnetic 4-vector potential.More generally, suppose the Lagrangian is that of a singleparticle plus an interaction term LI
L = −mc2dτ
dt+ LI .
Varying this with respect to the position of the particle rαas a function of time t gives
2.11. APPLICATIONS OR EXTENSIONS OF LAGRANGIAN MECHANICS IN OTHER CONTEXTS 25
δL = mdt
2dτδ
(gµν
dxµ
dt
dxν
dt
)+ δLI
= mdt
2dτ
(gµν,αδx
α dxµ
dt
dxν
dt+ 2gαν
dδxα
dt
dxν
dt
)+
∂LI
∂xαδxα +
∂LI
∂ dxα
dt
dδxα
dt
=1
2mgµν,αδx
α dxµ
dτ
dxν
dt− d
dt
(mgαν
dxν
dτ
)δxα +
∂LI
∂xαδxα − d
dt
(∂LI
∂ dxα
dt
)δxα +
d(...)
dt.
This gives the equation of motion
0 =1
2mgµν,α
dxµ
dτ
dxν
dt− d
dt
(mgαν
dxν
dτ
)+ fα
where
fα =∂LI
∂xα− d
dt
(∂LI
∂ dxα
dt
)
is the non-gravitational force on the particle. (For m tobe independent of time, we must have fα dxα
dt = 0 .)Rearranging gets the force equation
d
dt
(mdxν
dτ
)= −mΓν
µσ
dxµ
dτ
dxσ
dt+ gναfα
where Γ is the Christoffel symbol which is the gravita-tional force field.If we let
pν = mdxν
dτ
be the (kinetic) linearmomentum for a particle withmass,then
dpν
dt= −Γν
µσpµ dx
σ
dt+ gναfα
and
dxν
dt=
pν
p0
hold even for a massless particle.
2.11 Applications or extensionsof Lagrangian mechanics inother contexts
2.11.1 Relation to other formulations ofclassical mechanics
The Hamiltonian, denoted by H, is obtained by perform-ing a Legendre transformation on the Lagrangian, whichintroduces new variables, canonically conjugate to theoriginal variables. This doubles the number of variables,but makes differential equations first order. The Hamil-tonian is the basis for an alternative formulation of clas-sical mechanics known as Hamiltonian mechanics. It isa particularly ubiquitous quantity in quantum mechanics(see Hamiltonian (quantum mechanics)). In the classicalview, time is an independent variable and qi (and dqi/dt)are dependent variables as is often seen in phase spaceexplanations of systems.Routhian mechanics is a hybrid formulation of La-grangian and Hamiltonian mechanics, which is not oftenused in practice but an efficient formulation for cyclic co-ordinates.
2.11.2 Applications in quantummechanics
In 1948, Feynman discovered the path integral formula-tion extending the principle of least action to quantummechanics for electrons and photons. In this formula-tion, particles travel every possible path between the ini-tial and final states; the probability of a specific finalstate is obtained by summing over all possible trajecto-ries leading to it. In the classical regime, the path integralformulation cleanly reproduces Hamilton’s principle, andFermat’s principle in optics.
2.11.3 Classical field theory
In Lagrangian mechanics, the generalized coordinatesform a discrete set of variables that define the configu-ration of a system. The continuum analogue for defininga field are field variables, say ϕ(r, t), which representssome density function varying with position and time.In classical field theory, the physical system is not a setof discrete particles, but rather a continuous field definedover a region of 3d space. Associated with the field is aLagrangian density
L(ϕ,∇ϕ, ∂ϕ/∂t, r, t)
defined in terms of the field and its space and time deriva-tives at a location r and time t. The Lagrangian is thenthe integral of the Lagrangian density over 3d space (seevolume integral):
L(t) =
∫L d3r
26 CHAPTER 2. LAGRANGIAN MECHANICS
where d3r is a 3d differential volume element, must beused instead. The Lagrangian is a function of time sincethe Lagrangian density has implicit space dependence viathe fields, and may explicit spatial dependence, but theseare removed in the integral, leaving only time in as thevariable for the Lagrangian.
2.11.4 Uses in Engineering
Circa 1963 Lagrangians were a general part of the engi-neering curriculum, but a quarter of a century later, evenwith the ascendency of dynamical systems, they weredropped as requirements for some engineering programs,and are generally considered to be the domain of theoret-ical dynamics. Circa 2003 this changed dramatically, andLagrangians are not only a required part of manyME andEE graduate-level curricula, but also find applications infinance, economics, and biology, mainly as the basis ofthe formulation of various path integral schemes to facili-tate the solution of parabolic partial differential equationsvia random walks.Circa 2013, Lagrangians find their way into hundredsof direct engineering solutions, including robotics, tur-bulent flow analysis (Lagrangian and Eulerian specifi-cation of the flow field), signal processing, microscopiccomponent contact and nanotechnology (superlinear con-vergent augmented Lagrangians), gyroscopic forcing anddissipation, semi-infinite supercomputing (which also in-volve Lagrange multipliers in the subfield of semi-infiniteprogramming), chemical engineering (specific heat linearLagrangian interpolation in reaction planning), civil en-gineering (dynamic analysis of traffic flows), optics engi-neering and design (Lagrangian and Hamiltonian optics)aerospace (Lagrangian interpolation), force stepping in-tegrators, and even airbag deployment (coupled Eulerian-Lagrangians as well as SELM—the stochastic EulerianLagrangian method).[61]
2.12 See also
• Fundamental lemma of the calculus of variations
• Canonical coordinates
• Functional derivative
• Generalized coordinates
• Hamiltonian mechanics
• Hamiltonian optics
• Lagrangian analysis (applications of Lagrangianmechanics)
• Lagrangian point
• Lagrangian system
• Non-autonomous mechanics
• Restricted three-body problem
• Plateau’s problem
2.13 Footnotes[1] Sometimes in this context the variational derivative de-
noted and defined asδ
δrk≡ ∂
∂rk− d
dt
∂
∂rkis used. Throughout this article only partial and totalderivatives are used.
[2] Here the virtual displacements are assumed reversible, itis possible for some systems to have non-reversible virtualdisplacements that violate this principle, see Udwadia–Kalaba equation.
[3] The Lagrangian also can be written explicitly for a rotatingframe. See Padmanabhan, 2000.
[4] The line element squared is the Lorentz invariant
c2dτ2 = ηαβdxαdxβ = c2dt2 − dr2 ,
which takes the same values in all inertial frames of refer-ence. Here ηαβ are the components of the Minkowskimetric tensor, dxα = (cdt, dr) = (cdt, dx, dy, dz) arethe components of the differential in four position, thesummation convention over the covariant and contravari-ant spacetime indices α and β is used, each index takes thevalue 0 for timelike components, and 1, 2, 3 for spacelikecomponents, and
dr2 ≡ dr · dr ≡ dx2 + dy2 + dz2
is a shorthand for the square differential of the particle’sposition coordinates. Dividing by c2dt2 allows the con-version to the lab coordinate time as follows,
dτ2
dt2=
1
c2ηαβ
dxα
dt
dxβ
dt= 1− 1
c2dr2dt2
=1
γ(r)2
so that
dτ =1
c
√ηαβ
dxα
dt
dxβ
dtdt =
dt
γ(r) .
2.14 Notes[1] Dvorak & Freistetter 2005, p. 24
[2] Haken 2006, p. 61
[3] Lanczos 1986, p. 43
[4] Menzel & Zatzkis 1960, p. 160
[5] Feynman
[6] Goldstien, Poole & Safko 2001, p. 35
2.15. REFERENCES 27
[7] http://williamhoover.info/Scans1990s/1995-10.pdf
[8] Hand & Finch 2008, p. 36–40
[9] Torby1984, p.270
[10] Torby 1984, p. 269
[11] Hand & Finch 2008, p. 60–61
[12] Hand & Finch 2008, p. 19
[13] Penrose 2007
[14] Schuam 1988, p. 156
[15] Synge & Schild 1949, p. 150–152
[16] Hand & Finch 2008, p. 44–45
[17] Hand & Finch 2008, p. 44–45
[18] Foster & Nightingale 1995, p. 89
[19] Hand & Finch 2008, p. 4
[20] Goldstein 1980, p. 16–18
[21] Hand 2008, p. 15
[22] Hand & Finch 2008, p. 15
[23] Fetter & Walecka 1980, p. 53
[24] Torby 1984, p. 264
[25] Torby 1984, p. 269
[26] Kibble & Berkshire 2004, p. 234
[27] Fetter & Walecka 1980, p. 56
[28] Hand & Finch 2008, p. 17
[29] Hand & Finch 2008, p. 15–17
[30] R. Penrose (2007). The Road to Reality. Vintage books.p. 474. ISBN 0-679-77631-1.
[31] Goldstien 1980, p. 23
[32] Kibble & Berkshire 2004, p. 234–235
[33] Hand & Finch 2008, p. 51
[34] Fetter Walecka, pp. 68–70
[35] Landau & Lifshitz 1976, p. 4
[36] Goldstien, Poole & Safko 2002, p. 21
[37] Landau & Lifshitz 1976, p. 4
[38] Goldstien 1980, p. 21
[39] Landau & Lifshitz 1976, p. 14
[40] Landau & Lifshitz 1976, p. 22
[41] Taylor 2005, p. 297
[42] Padmanabhan 2000, p. 48
[43] Hand & Finch 1998, pp. 140–141
[44] Hildebrand 1992, p. 156
[45] Zak, Zbilut & Meyers 1997, pp. 202
[46] Shabana 2008, pp. 118–119
[47] Gannon 2006, p. 267
[48] Kosyakov 2007
[49] Galley 2013
[50] Hadar, Shahar & Kol 2014
[51] Birnholtz, Hadar & Kol 2013
[52] Torby 1984, p. 271
[53] Goldstein 1980, p. 328
[54] Hobson, Efstathiou & Lasenby 2006, p. 79–80
[55] Foster & Nightingale 1995, p. 62–63
[56] Hobson, Efstathiou & Lasenby 2006, p. 79–80
[57] Goldstein 1980, p. 324
[58] Hand & Finch 2008, p. 551
[59] Goldstein 1980, p. 323
[60] Hand & Finch 2008, p. 534
[61] Gans 2013
2.15 References• Penrose, Roger (2007). The Road to Reality. Vin-tage books. ISBN 0-679-77631-1.
• Landau, L. D.; Lifshitz, E. M.. Mechanics (3rded.). Butterworth Heinemann. p. 134. ISBN9780750628969.
• Landau, Lev; Lifshitz, Evgeny (1975). The ClassicalTheory of Fields. Elsevier Ltd. ISBN 978-0-7506-2768-9.
• Hand, L. N.; Finch, J. D. Analytical Mechanics (2nded.). Cambridge University Press. p. 23. ISBN9780521575720.
• Louis N. Hand, Janet D. Finch (1998). Analyticalmechanics. Cambridge University Press. pp. 140–141. ISBN 0-521-57572-9.
• Kibble, T. W. B.; Berkshire, F. H. (2004). Classi-cal Mechanics (5th ed.). Imperial College Press. p.236. ISBN 9781860944352.
• Goldstein, Herbert (1980). Classical Mechanics(2nd ed.). San Francisco, CA: Addison Wesley. pp.352–353. ISBN 0201029189.
28 CHAPTER 2. LAGRANGIAN MECHANICS
• Goldstein, Herbert; Poole, Charles P., Jr.; Safko,John L. (2002). Classical Mechanics (3rd ed.). SanFrancisco, CA: Addison Wesley. pp. 347–349.ISBN 0-201-65702-3.
• Lanczos, Cornelius (1986). “II §5 Auxiliary condi-tions: the Lagrangian λ-method”. The variationalprinciples of mechanics (Reprint of University ofToronto 1970 4th ed.). Courier Dover. p. 43. ISBN0-486-65067-7.
• Fetter, A. L.; Walecka, J. D. (1980). TheoreticalMechanics of Particles and Continua. Dover. pp.53–57. ISBN 978-0-486-43261-8.
• The Principle of Least Action, R. Feynman
• Dvorak, R.; Freistetter, Florian (2005). "§ 3.2 La-grange equations of the first kind”. Chaos and sta-bility in planetary systems. Birkhäuser. p. 24. ISBN3-540-28208-4.
• Haken, H (2006). Information and self-organization(3rd ed.). Springer. p. 61. ISBN 3-540-33021-6.
• Henry Zatzkis (1960). "§1.4 Lagrange equations ofthe second kind”. In DHMenzel. Fundamental for-mulas of physics 1 (2nd ed.). Courier Dover. p. 160.ISBN 0-486-60595-7.
• Francis Begnaud Hildebrand (1992). Methods ofapplied mathematics (Reprint of Prentice-Hall 19652nd ed.). Courier Dover. p. 156. ISBN 0-486-67002-3.
• Michail Zak, Joseph P. Zbilut, Ronald E. Meyers(1997). From instability to intelligence. Springer. p.202. ISBN 3-540-63055-4.
• Ahmed A. Shabana (2008). Computational contin-uum mechanics. Cambridge University Press. pp.118–119. ISBN 0-521-88569-8.
• John Robert Taylor (2005). Classical mechanics.University Science Books. p. 297. ISBN 1-891389-22-X.
• Padmanabhan, Thanu (2000). "§2.3.2 Motion ina rotating frame”. Theoretical Astrophysics: Astro-physical processes (3rd ed.). Cambridge UniversityPress. p. 48. ISBN 0-521-56632-0.
• Doughty, Noel A. (1990). Lagrangian Interac-tion. Addison-Wesley Publishers Ltd. ISBN 0-201-41625-5.
• Kosyakov, B. P. (2007). Introduction to the classi-cal theory of particles and fields. Berlin, Germany:Springer. doi:10.1007/978-3-540-40934-2.
• Galley, Chad R. (2013). “Classical Mechanicsof Nonconservative Systems”. Physical ReviewLetters 110 (17): 174301. arXiv:1210.2745.Bibcode:2013PhRvL.110q4301G.doi:10.1103/PhysRevLett.110.174301. PMID23679733.
• Birnholtz, Ofek; Hadar, Shahar; Kol, Barak(2014). “Radiation reaction at the level ofthe action”. International Journal of ModernPhysics A 29 (24): 1450132. arXiv:1402.2610.Bibcode:2014IJMPA..2950132B.doi:10.1142/S0217751X14501322.
• Birnholtz, Ofek; Hadar, Shahar; Kol, Barak(2013). “Theory of post-Newtonian ra-diation and reaction”. Physical ReviewD 88 (10): 104037. arXiv:1305.6930.Bibcode:2013PhRvD..88j4037B.doi:10.1103/PhysRevD.88.104037.
• Roger F Gans (2013). Engineering Dynamics: Fromthe Lagrangian to Simulation. New York: Springer.ISBN 978-1-4614-3929-5.
• Terry Gannon (2006). Moonshine beyond the mon-ster: the bridge connecting algebra, modular formsand physics. Cambridge University Press. p. 267.ISBN 0-521-83531-3.
• Torby, Bruce (1984). “EnergyMethods”. AdvancedDynamics for Engineers. HRW Series in Mechani-cal Engineering. United States of America: CBSCollege Publishing. ISBN 0-03-063366-4.
• Foster, J; Nightingale, J.D. (1995). A Short Coursein General Relativity (2nd ed.). Springer. ISBN 0-03-063366-4.
• M. P. Hobson, G. P. Efstathiou, A. N. Lasenby(2006). General Relativity: An Introduction forPhysicists. Cambridge University Press. p. 79–80.ISBN 9780521829519.
2.16 Further reading• Gupta, Kiran Chandra, Classical mechanics of par-ticles and rigid bodies (Wiley, 1988).
• Cassel, Kevin W.: Variational Methods with Appli-cations in Science and Engineering, Cambridge Uni-versity Press, 2013.
2.17. EXTERNAL LINKS 29
2.17 External links• Tong, David, Classical Dynamics Cambridge lecturenotes
• Principle of least action interactive Excellent inter-active explanation/webpage
• Joseph Louis de Lagrange - Œuvres complètes(Gallica-Math)
Chapter 3
Action (physics)
In physics, action is an attribute of the dynamics of aphysical system. It is a mathematical functional whichtakes the trajectory, also called path or history, of the sys-tem as its argument and has a real number as its result.Generally, the action takes different values for differentpaths.[1] Action has the dimensions of [energy]·[time],and its SI unit is joule-second. This is the same unit asthat of angular momentum.
3.1 Introduction
Empirical laws are frequently expressed as differentialequations, which describe how physical quantities such asposition and momentum change continuously with time.Given the initial and boundary conditions for the situa-tion, the “solution” to these empirical equations is an im-plicit function describing the behavior of the system.There is an alternative approach to finding equations ofmotion. Classical mechanics postulates that the path ac-tually followed by a physical system is that for which theaction is minimized, or, more generally, is stationary. Inother words, the action satisfies a variational principle:the principle of stationary action (see also below). Theaction is defined by an integral, and the classical equa-tions of motion of a system can be derived by minimizingthe value of that integral.This simple principle provides deep insights into physics,and is an important concept in modern theoreticalphysics.The equivalence of these two approaches is containedin Hamilton’s principle, which states that the differentialequations of motion for any physical system can be re-formulated as an equivalent integral equation. It appliesnot only to the classical mechanics of a single particle,but also to classical fields such as the electromagnetic andgravitational fields. Hamilton’s principle has also beenextended to quantum mechanics and quantum field the-ory—in particular path integral formulation makes use ofthe concept—where a physical system follows simultane-ously all possible paths with probability amplitudes foreach path being determined by the action for the path.[2]
3.2 History
Action was defined in several, now obsolete, ways duringthe development of the concept.[3]
• Gottfried Leibniz, Johann Bernoulli and PierreLouis Maupertuis defined the action for light as theintegral of its speed or inverse speed along its pathlength.
• Leonhard Euler (and, possibly, Leibniz) defined ac-tion for a material particle as the integral of the par-ticle’s speed along its path through space.
• Pierre Louis Maupertuis introduced several ad hocand contradictory definitions of action within a sin-gle article, defining action as potential energy, as vir-tual kinetic energy, and as a hybrid that ensured con-servation of momentum in collisions.[4]
3.3 Mathematical definition
Expressed in mathematical language, using the calculusof variations, the evolution of a physical system (i.e., howthe system actually progresses from one state to another)corresponds to a stationary point (usually, a minimum) ofthe action.Several different definitions of 'the action' are in commonuse in physics.[3][5] The action is usually an integral overtime. But for action pertaining to fields, it may be inte-grated over spatial variables as well. In some cases, theaction is integrated along the path followed by the physi-cal system.The action is typically represented as an integral overtime, taken along the path of the system between the ini-tial time and the final time of the development of thesystem,[3]
S =
∫ t2
t1
Ldt ,
where the integrand L is called the Lagrangian. For the
30
3.4. ACTION IN CLASSICAL PHYSICS (DISAMBIGUATION) 31
action integral to be well-defined the trajectory has to bebounded in time and space.Action has the dimensions of [energy]·[time], and its SIunit is joule-second. Dimensionally, action has the sameunits as angular momentum.
3.4 Action in classical physics (dis-ambiguation)
In classical physics, the term “action” has a number ofmeanings.
3.4.1 Action (functional)
Most commonly, the term is used for a functionalS whichtakes a function of time and (for fields) space as inputand returns a scalar.[6][7] In classical mechanics, the inputfunction is the evolution q(t) of the system between twotimes t1 and t2, where q represent the generalized coor-dinates. The action S[q(t)] is defined as the integral ofthe Lagrangian L for an input evolution between the twotimes
S[q(t)] =∫ t2
t1
L[q(t), q(t), t] dt
where the endpoints of the evolution are fixed and de-fined as q1 = q(t1) and q2 = q(t2) . According toHamilton’s principle, the true evolution q ᵣᵤₑ(t) is an evo-lution for which the action S[q(t)] is stationary (a mini-mum, maximum, or a saddle point). This principle resultsin the equations of motion in Lagrangian mechanics.
3.4.2 Abbreviated action (functional)
Usually denoted as S0 , this is also a functional. Here theinput function is the path followed by the physical systemwithout regard to its parameterization by time. For exam-ple, the path of a planetary orbit is an ellipse, and the pathof a particle in a uniform gravitational field is a parabola;in both cases, the path does not depend on how fast theparticle traverses the path. The abbreviated action S0 isdefined as the integral of the generalized momenta alonga path in the generalized coordinates
S0 =
∫p · dq =
∫pi dqi
According toMaupertuis’ principle, the true path is a pathfor which the abbreviated action S0 is stationary.
3.4.3 Hamilton’s principal function
Main article: Hamilton’s principal function
Hamilton’s principal function is defined by the Hamilton–Jacobi equations (HJE), another alternative formulationof classical mechanics. This function S is related to thefunctional S by fixing the initial time t1 and endpointq1 and allowing the upper limits t2 and the second end-point q2 to vary; these variables are the arguments of thefunction S. In other words, the action function S is theindefinite integral of the Lagrangian with respect to time.
3.4.4 Hamilton’s characteristic function
When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separationof variables
S(q1, . . . , qN , t) = W (q1, . . . , qN )− E · t
where the time independent function W(q1, q2 ... qN)is called Hamilton’s characteristic function. The physicalsignificance of this function is understood by taking itstotal time derivative
dW
dt=
∂W
∂qiqi = piqi
This can be integrated to give
W (q1, . . . , qN ) =
∫piqi dt =
∫pi dqi
which is just the abbreviated action.
3.4.5 Other solutions of Hamilton–Jacobiequations
The Hamilton–Jacobi equations are often solved by ad-ditive separability; in some cases, the individual terms ofthe solution, e.g., Sk(qk), are also called an “action”.[3]
3.4.6 Action of a generalized coordinate
This is a single variable Jk in the action-angle coordi-nates, defined by integrating a single generalized momen-tum around a closed path in phase space, correspondingto rotating or oscillating motion
Jk =
∮pkdqk
32 CHAPTER 3. ACTION (PHYSICS)
The variable Jk is called the “action” of the generalizedcoordinate qk; the corresponding canonical variable con-jugate to Jk is its “angle” wk, for reasons described morefully under action-angle coordinates. The integration isonly over a single variable qk and, therefore, unlike theintegrated dot product in the abbreviated action integralabove. The Jk variable equals the change in Sk(qk) asqk is varied around the closed path. For several physicalsystems of interest, J is either a constant or varies veryslowly; hence, the variable Jk is often used in perturbationcalculations and in determining adiabatic invariants.
3.4.7 Action for a Hamiltonian flow
See tautological one-form.
3.5 Euler–Lagrange equations forthe action integral
As noted above, the requirement that the action integralbe stationary under small perturbations of the evolutionis equivalent to a set of differential equations (called theEuler–Lagrange equations) that may be determined us-ing the calculus of variations. We illustrate this deriva-tion here using only one coordinate, x; the extension tomultiple coordinates is straightforward.[1][7]
Adopting Hamilton’s principle, we assume that the La-grangian L (the integrand of the action integral) dependsonly on the coordinate x(t) and its time derivative dx(t)/dt,and may also depend explicitly on time. In that case, theaction integral can be written
S =
∫ t2
t1
L(x, x, t) dt
where the initial and final times (t1 and t2) and the fi-nal and initial positions are specified in advance as x1 =x(t1) and x2 = x(t2) . Let x ᵣᵤₑ(t) represent the true evo-lution that we seek, and let xper(t) be a slightly perturbedversion of it, albeit with the same endpoints, xper(t1) =x1 and xper(t2) = x2 . The difference between thesetwo evolutions, which we will call ε(t) , is infinitesimallysmall at all times
ε(t) = xper(t)− xtrue(t)
At the endpoints, the difference vanishes, i.e., ε(t1) =ε(t2) = 0 .Expanded to first order, the difference between the ac-tions integrals for the two evolutions is
δS =
∫ t2
t1
[L(xtrue + ε, xtrue + ε, t)− L(xtrue, xtrue, t)] dt
=
∫ t2
t1
(ε∂L
∂x+ ε
∂L
∂x
)dt
Integration by parts of the last term, together with theboundary conditions ε(t1) = ε(t2) = 0 , yields the equa-tion
δS =
∫ t2
t1
(ε∂L
∂x− ε
d
dt
∂L
∂x
)dt.
The requirement thatS be stationary implies that the first-order change must be zero for any possible perturbationε(t) about the true evolution,
This can be true only if
The Euler–Lagrange equation is obeyed provided thefunctional derivative of the action integral is identicallyzero:
δSδx(t)
= 0
The quantity ∂L∂x is called the conjugate momentum for the
coordinate x. An important consequence of the Euler–Lagrange equations is that if L does not explicitly containcoordinate x, i.e.
if ∂L∂x = 0 , then ∂L
∂x is constant in time.
In such cases, the coordinate x is called a cyclic coordi-nate, and its conjugate momentum is conserved.
3.5.1 Example: free particle in polar coor-dinates
Simple examples help to appreciate the use of the actionprinciple via the Euler–Lagrangian equations. A free par-ticle (mass m and velocity v) in Euclidean space moves ina straight line. Using the Euler–Lagrange equations, thiscan be shown in polar coordinates as follows. In the ab-sence of a potential, the Lagrangian is simply equal to thekinetic energy
L =1
2mv2 =
1
2m(x2 + y2
)
3.6. THE ACTION PRINCIPLE 33
in orthonormal (x,y) coordinates, where the dot repre-sents differentiation with respect to the curve parameter(usually the time, t). In polar coordinates (r, φ) the ki-netic energy and hence the Lagrangian becomes
L =1
2m(r2 + r2φ2
).
The radial r and φ components of the Euler–Lagrangianequations become, respectively
d
dt
(∂L
∂r
)− ∂L
∂r= 0 ⇒ r − rφ2 = 0
d
dt
(∂L
∂φ
)− ∂L
∂φ= 0 ⇒ φ+
2
rrφ = 0
The solution of these two equations is given by
r cosφ = at+ b
r sinφ = ct+ d
for a set of constants a, b, c, d determined by initial con-ditions. Thus, indeed, the solution is a straight line givenin polar coordinates.
3.6 The action principle
Main article: principle of stationary action
3.6.1 Classical fields
See also: Einstein–Hilbert action
The action principle can be extended to obtain theequations ofmotion for fields, such as the electromagneticfield or gravitational field.The Einstein equation utilizes the Einstein–Hilbert actionas constrained by a variational principle.The trajectory (path in spacetime) of a body in a gravita-tional field can be found using the action principle. For afree falling body, this trajectory is a geodesic.
3.6.2 Conservation laws
Main article: Conservation laws
Implications of symmetries in a physical situation can befound with the action principle, together with the Euler–Lagrange equations, which are derived from the actionprinciple. An example is Noether’s theorem, which states
that to every continuous symmetry in a physical situationthere corresponds a conservation law (and conversely).This deep connection requires that the action principlebe assumed.[2]
3.6.3 Quantum mechanics and quantumfield theory
Main articles: Quantum mechanics and quantum fieldtheory
In quantum mechanics, the system does not follow a sin-gle path whose action is stationary, but the behavior ofthe system depends on all permitted paths and the valueof their action. The action corresponding to the variouspaths is used to calculate the path integral, that gives theprobability amplitudes of the various outcomes.Although equivalent in classical mechanics with Newton’slaws, the action principle is better suited for generaliza-tions and plays an important role in modern physics. In-deed, this principle is one of the great generalizations inphysical science. It is best understood within quantummechanics. In particular, in Richard Feynman's path in-tegral formulation of quantum mechanics, where it arisesout of destructive interference of quantum amplitudes.Maxwell’s equations can also be derived as conditions ofstationary action.
3.6.4 Single relativistic particle
Main article: Theory of relativity
When relativistic effects are significant, the action ofa point particle of mass m travelling a world line Cparametrized by the proper time τ is
S = −mc2∫C
dτ
If instead, the particle is parametrized by the coordinatetime t of the particle and the coordinate time ranges fromt1 to t2, then the action becomes
∫ t2
t1
Ldt
where the Lagrangian is
L = −mc2√
1− v2
c2 .[8]
3.6.5 Modern extensions
The action principle can be generalized still further. Forexample, the action need not be an integral because
34 CHAPTER 3. ACTION (PHYSICS)
nonlocal actions are possible. The configuration spaceneed not even be a functional space given certain featuressuch as noncommutative geometry. However, a physicalbasis for these mathematical extensions remains to be es-tablished experimentally.[6]
3.7 See also• Calculus of variations
• Functional derivative
• Functional integral
• Hamiltonian mechanics
• Lagrangian
• Lagrangian mechanics
• Measure (physics)
• Noether’s theorem
• Path integral formulation
• Planck’s constant
• Principle of least action
• Quantum physics
• Entropy (the least Action Principle and the Principleof Maximum Probability or Entropy could be seenas analogous)
3.8 References[1] McGraw Hill Encyclopaedia of Physics (2nd Edition),
C.B. Parker, 1994, ISBN 0-07-051400-3
[2] Quantum Mechanics, E. Abers, Pearson Ed., AddisonWesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
[3] Analytical Mechanics, L.N. Hand, J.D. Finch, CambridgeUniversity Press, 2008, ISBN 978-0-521-57572-0
[4] Œuvres deMr deMaupertuis (pre-1801 Imprint Collectionat the Library of Congress).
[5] Encyclopaedia of Physics (2nd Edition), R.G. Lerner,G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsge-sellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
[6] The Road to Reality, Roger Penrose, Vintage books,2007, ISBN 0-679-77631-1
[7] Classical Mechanics, T.W.B. Kibble, European PhysicsSeries, McGraw-Hill (UK), 1973, ISBN 0-07-084018-0
[8] L.D. Landau and E.M. Lifshitz The Classical Theory ofFields Addison-Wesley 1971 sec 8.p.24-25
3.9 Sources and further reading
For an annotated bibliography, see Edwin F. Taylor wholists, among other things, the following books
• The Cambridge Handbook of Physics Formulas, G.Woan, Cambridge University Press, 2010, ISBN978-0-521-57507-2.
• Cornelius Lanczos, The Variational Principles ofMechanics (Dover Publications, New York, 1986).ISBN 0-486-65067-7. The reference most quotedby all those who explore this field.
• L. D. Landau and E.M. Lifshitz, Mechanics, Courseof Theoretical Physics (Butterworth-Heinenann,1976), 3rd ed., Vol. 1. ISBN 0-7506-2896-0. Be-gins with the principle of least action.
• Thomas A. Moore “Least-Action Principle” inMacmillan Encyclopedia of Physics (Simon &Schuster Macmillan, 1996), Volume 2, ISBN 0-02-897359-3, OCLC 35269891, pages 840 – 842.
• Gerald Jay Sussman and Jack Wisdom, Structureand Interpretation of Classical Mechanics (MITPress, 2001). Begins with the principle of leastaction, uses modern mathematical notation, andchecks the clarity and consistency of procedures byprogramming them in computer language.
• Dare A. Wells, Lagrangian Dynamics, Schaum’sOutline Series (McGraw-Hill, 1967) ISBN 0-07-069258-0, A 350-page comprehensive “outline” ofthe subject.
• Robert Weinstock, Calculus of Variations, with Ap-plications to Physics and Engineering (Dover Publi-cations, 1974). ISBN 0-486-63069-2. An oldie butgoodie, with the formalism carefully defined beforeuse in physics and engineering.
• Wolfgang Yourgrau and StanleyMandelstam, Varia-tional Principles in Dynamics and Quantum Theory(Dover Publications, 1979). A nice treatment thatdoes not avoid the philosophical implications of thetheory and lauds the Feynman treatment of quantummechanics that reduces to the principle of least ac-tion in the limit of large mass.
• Edwin F. Taylor’s page
• Principle of least action interactive Excellent inter-active explanation/webpage
3.10 External links
Chapter 4
AQUAL
AQUAL is a theory of gravity based onModified Newto-nian Dynamics (MOND), but using a Lagrangian. It wasdeveloped by Jacob Bekenstein andMordehaiMilgrom intheir 1984 paper, “Does the missing mass problem sig-nal the breakdown of Newtonian gravity?". “AQUAL”stands for “A QUAdratic Lagrangian”.The gravitational force law obtained from MOND,
mµ(a/a0)a = GMm/r2
has a serious defect: it violates Newton’s third law of mo-tion, and therefore fails to conserve momentum and en-ergy. To see this, consider two objects with m = M ;then we have
µ(am/a0)mam = GmM/r2 = GMm/r2 = µ(aM/a0)MaM
but the third law gives mam = MaM , so we would getµ(am/a0) = µ(aM/a0) even though am = aM , andµ would therefore be constant, contrary to the MONDassumption that it is linear for small arguments.This problem can be rectified by deriving the force lawfrom a Lagrangian, at the cost of possibly modifying thegeneral form of the force law. Then conservation lawscould then be derived from the Lagrangian by the usualmeans.The AQUAL Lagrangian is:
ρΦ+ (8πG)−1a20F (|∇Φ|2/a20)
this leads to a modified Poisson equation:
∇ · (µ(|∇Φ|/a0)∇Φ) = 4πGρ
Here, µ(x) = dF (x2)/dx and the predicted acceler-ation is −∇Φ = a . These equations reduce to theMOND equations in the spherically symmetric case, al-though they differ somewhat in the disc case needed formodelling spiral or lenticular galaxies. However, the dif-ference is only 10-15%, so does not seriously impact theresults.
4.1 References• Jacob Bekenstein and M. Milgrom (1984).“Does the missing mass problem signal thebreakdown of Newtonian gravity?". Astrophys.J. 286: 7–14. Bibcode:1984ApJ...286....7B.doi:10.1086/162570.
• Milgrom, M (1986). “Solutions for the modifiedNewtonian dynamics field equation”. Astrophys.J. 302: 617–625. Bibcode:1986ApJ...302..617M.doi:10.1086/164021.
• Bekenstein, Jacob D (2009). “Relativistic MONDas an alternative to the dark matter paradigm”. Nu-clear Physics A 827: 555c–560c. arXiv:0901.1524.Bibcode:2009NuPhA.827..555B.doi:10.1016/j.nuclphysa.2009.05.122.
35
Chapter 5
Averaged Lagrangian
High-altitude wave cloud formed over the Hampton area atBurra, South Australia on 16 January 2007.
In continuum mechanics, Whitham’s averaged La-grangian method – or in shortWhitham’s method – isused to study the Lagrangian dynamics of slowly-varyingwave trains in an inhomogeneous (moving) medium. Themethod is applicable to both linear and non-linear sys-tems. As a direct consequence of the averaging used inthe method, wave action is a conserved property of thewave motion. In contrast, the wave energy is not nec-essarily conserved, due to the exchange of energy withthe mean motion. However the total energy, the sum ofthe energies in the wave motion and the mean motion,will be conserved for a time-invariant Lagrangian. Fur-ther, the averaged Lagrangian has a strong relation to thedispersion relation of the system.The method is due to Gerald Whitham, who developedit in the 1960s. It is for instance used in the modellingof surface gravity waves on fluid interfaces,[1][2] and inplasma physics.[3][4]
5.1 Resulting equations for purewave motion
In case a Lagrangian formulation of a continuum me-chanics system is available, the averaged Lagrangianmethodology can be used to find approximations for theaverage dynamics of wave motion – and (eventually) forthe interaction between the wave motion and the mean
motion – assuming the envelope dynamics of the car-rier waves is slowly varying. Phase averaging of the La-grangian results in an averaged Lagrangian, which is al-ways independent of the wave phase itself (but dependson slowly varying wave quantities like wave amplitude,frequency and wavenumber). By Noether’s theorem,variation of the averaged Lagrangian L with respect tothe invariant wave phase θ(x, t) then gives rise to aconservation law:[5]
This equation states the conservation of wave action – ageneralization of the concept of an adiabatic invariant tocontinuum mechanics – with[6]
A ≡ − ∂L∂(∂tθ)
= +∂L∂ω and B ≡ − ∂L
∂(∇θ) =
−∂L∂k
being the wave action A and wave action flux B respec-tively. Further x and t denote space and time respec-tively, while∇ is the gradient operator. The angular fre-quency ω(x, t) and wavenumber k(x, t) are defined as[7]
and both are assumed to be slowly varying. Due to thisdefinition, ω(x, t) and k(x, t) have to satisfy the consis-tency relations:
The first consistency equation is known as the conser-vation of wave crests, and the second states that thewavenumber field k(x, t) is irrotational (i.e. has zerocurl).
5.2 Method
The averaged Lagrangian approach applies to wave mo-tion – possibly superposed on a mean motion – that canbe described in a Lagrangian formulation. Using an
36
5.2. METHOD 37
ansatz on the form of the wave part of the motion, theLagrangian is phase averaged. Since the Lagrangian isassociated with the kinetic energy and potential energy ofthe motion, the oscillations contribute to the Lagrangian,although the mean value of the wave’s oscillatory excur-sion is zero (or very small).The resulting averaged Lagrangian contains wave char-acteristics like the wavenumber, angular frequency andamplitude (or equivalently the wave’s energy density orwave action). But the wave phase itself is absent dueto the phase averaging. Consequently, through Noether’stheorem, there is a conservation law called the conserva-tion of wave action.Originally the averaged Lagrangian method was de-veloped by Whitham for slowly-varying dispersivewave trains.[8] Several extensions have been made,e.g. to interacting wave components,[9][10] Hamiltonianmechanics,[8][11] higher-order modulational effects,[12]dissipation effects.[13]
5.2.1 Variational formulation
The averaged Lagrangian method requires the existenceof a Lagrangian describing the wave motion. For instancefor a field φ(x, t) , described by a Lagrangian densityL (∂tφ,∇φ,φ) , the principle of stationary action is:[14]
δ
(∫ ∫∫∫L (∂tφ(x, t),∇φ(x, t), φ(x, t)) dx dt
)= 0,
with ∇ the gradient operator and ∂t the time deriva-tive operator. This action principle results in the Euler–Lagrange equation:[14]
∂t
(∂L
∂ (∂tφ)
)+∇ ·
(∂L
∂ (∇φ)
)− ∂L
∂φ= 0,
which is the second-order partial differential equation de-scribing the dynamics of φ. Higher-order partial differ-ential equations require the inclusion of higher than first-order derivatives in the Lagrangian.[14]
Example
For example, consider a non-dimensional and non-linearKlein–Gordon equation in one space dimension x :[15]
This Euler–Lagrange equation emerges from the La-grangian density:[15]
The small-amplitude approximation for the Sine–Gordonequation corresponds with the value σ = − 1
24 .[16]
For σ = 0 the system is linear and the classical one-dimensional Klein–Gordon equation is obtained.
5.2.2 Slowly-varying waves
Slowly-varying linear waves
Whitham developed several approaches to obtain a aver-aged Lagrangian method.[14][17] The simplest one is forslowly-varying linear wavetrains, which method will beapplied here.[14]
The slowly-varying wavetrain – without mean motion –in a linear dispersive system is described as:[18]
φ ∼ ℜA eiθ
= a cos (θ + α) , with a =
|A| and α = arg A ,
where θ is the real-valued wave phase, |A| denotes theabsolute value of the complex-valued amplitudeA, whileargA is its argument and ℜA denotes its real part.The real-valued amplitude and phase shift are denoted bya and α respectively.Now, by definition, the angular frequency ω andwavenumber vector k are expressed as the time deriva-tive and gradient of the wave phase θ(x, t) as:[7]
ω ≡ −∂tθ and k ≡ +∇θ.
As a consequence, ω(x, t) and k(x, t) have to satisfy theconsistency relations:
∂tk +∇ω = 0 and∇× k = 0.
These two consistency relations denote the “conservationof wave crests”, and the irrotationality of the wavenumberfield.Because of the assumption of slow variations in the wavetrain – as well as in a possible inhomogeneous mediumandmeanmotion – the quantitiesA, a, ω,k andα all varyslowly in space x and time t – but the wave phase θ itselfdoes not vary slowly. Consequently, derivatives of a, ω, kandα are neglected in the determination of the derivativesof φ(x, t) for use in the averaged Lagrangian:[14]
∂tφ ≈ +ω a sin(θ + α) and ∇φ ≈−k a sin(θ + α).
Next these assumptions on φ(x, t) and its derivatives areapplied to the Lagrangian density L (∂tφ,∇φ,φ) .
Slowly-varying non-linear waves
Several approaches to slowly-varying non-linear wave-trains are possible. One is by the use of Stokes ex-pansions,[19] used by Whitham to analyse slowly-varyingStokes waves.[20] A Stokes expansion of the field φ(x, t)can be written as:[19]
38 CHAPTER 5. AVERAGED LAGRANGIAN
φ = a cos (θ + α)+a2 cos (2θ + α2)+a3 cos (3θ + α3)+· · · ,
where the amplitudes a, a2, etc. are slowly varying, as arethe phases α, α2, etc. As for the linear wave case, in low-est order (as far as modulational effects are concerned)derivatives of amplitudes and phases are neglected, ex-cept for derivatives ω and k of the fast phase θ :
∂tφ ≈ +ωa sin (θ + α) +2ωa2 sin (2θ + α2) + 3ωa3 sin (3θ + α3) +· · · , and∇φ ≈ −ka sin (θ + α) −2ka2 sin (2θ + α2) − 3ka3 sin (3θ + α3) +· · · .
These approximations are to be applied in the Lagrangiandensity L , and its phase average L.
5.2.3 Averaged Lagrangian for slowly-varying waves
For pure wave motion the Lagrangian L (∂tφ,∇φ,φ)is expressed in terms of the field φ(x, t) and itsderivatives.[14][17] In the averaged Lagrangian method,the above-given assumptions on the field φ(x, t) – and itsderivatives – are applied to calculate the Lagrangian. TheLagrangian is thereafter averaged over the wave phase θ :[14]
L =1
2π
∫ 2π
0
L (∂tφ,∇φ,φ) dθ.
As a last step, this averaging result L can be expressed asthe averaged Lagrangian density L(ω,k, a) – which is afunction of the slowly varying parameters ω, k and a andindependent of the wave phase θ itself.[14]
The averaged Lagrangian density L is now proposed byWhitham to follow the average variational principle:[14]
From the variations of L follow the dynamical equationsfor the slowly-varying wave properties.
Example
Continuing on the example of the nonlinear Klein–Gordon equation, see equations 4 and 5, and applying theabove approximations for φ, ∂tφ and ∂xφ (for this 1Dexample) in the Lagrangian density, the result after aver-aging over θ is:
L = 14 (ω
2−k2−1)a2− 332σa
4+(ω2−k2− 14 )a
22+O(a6),
where it has been assumed that, in big-O notation, a2 =O(a2) and a3 = O(a3) . Variation of L with respect toa2 leads to a2 = 0. So the averaged Lagrangian is:
For linear wave motion the averaged Lagrangian is ob-tained by setting σ equal to zero.
5.2.4 Set of equations emerging from theaveraged Lagrangian
Applying the averaged Lagrangian principle, variationwith respect to the wave phase θ leads to the conserva-tion of wave action:
∂t
(+∂L∂ω
)+∇ ·
(−∂L∂k
)= 0,
since ω = −∂tθ and k = ∇θ while the wave phase θdoes not appear in the averaged Lagrangian densityL dueto the phase averaging. Defining the wave action as A ≡+∂L/∂ω and the wave action flux as B ≡ −∂L/∂k theresult is:
∂tA+∇ ·B = 0.
The wave action equation is accompanied with the con-sistency equations for ω and k which are:
∂tk +∇ω = 0 and∇× k = 0.
Variation with respect to the amplitude a leads to thedispersion relation ∂L/∂a = 0.
Example
Continuing with the nonlinear Klein–Gordon equation,using the average variational principle on equation 6, thewave action equation becomes by variation with respectto the wave phase θ :
∂t(12ωa
2)+ ∂x
(12ka
2)= 0,
and the nonlinear dispersion relation follows from varia-tion with respect to the amplitude a :
ω2 = k2 + 1 + 34σa
2.
So the wave action isA = 12ωa
2 and the wave action fluxB = 1
2ka2. The group velocity vg is vg ≡ B/A = k/ω.
5.6. REFERENCES 39
5.2.5 Mean motion and pseudo-phase
5.3 Conservation of wave action
The averaged Lagrangian is obtained by integration of theLagrangian over the wave phase. As a result, the aver-aged Lagrangian only contains the derivatives of the wavephase θ (these derivatives being, by definition, the angu-lar frequency and wavenumber) and does not depend onthe wave phase itself. So the solutions will be indepen-dent of the choice of the zero level for the wave phase.Consequently – by Noether’s theorem – variation of theaveraged Lagrangian L with respect to the wave phaseresults in a conservation law:
where
A ≡ δLδω
= − δLδ (∂tθ)
and B ≡ −δLδk
=
− δLδ (∇θ)
,
with A the wave action and B the wave action flux. Fur-ther ∂t denotes the partial derivative with respect to time,and ∇ is the gradient operator. By definition, the groupvelocity vg is given by:
B ≡ vgA.
Note that in general the energy of the wave motion doesnot need to be conserved, since there can be an energyexchange with a mean flow. The total energy – the sumof the energies of the wave motion and the mean flow – isconserved (when there is no work by external forces andno energy dissipation by viscosity).Conservation of wave action is also found by applyingthe generalized Lagrangian mean (GLM) method to theequations of the combined flow of waves and mean mo-tion, using Newtonian mechanics instead of a variationalapproach.[21]
5.4 Conservation of energy andmomentum
5.5 Connection to the dispersionrelation
Pure wave motion by linear models always leads to an av-eraged Lagrangian density of the form:[14]
L = G(ω,k)a2.
Consequently, the variation with respect to amplitude:∂L/∂a = 0 gives
G(ω,k) = 0.
So this turns out to be the dispersion relation for the lin-ear waves, and the averaged Lagrangian for linear wavesis always the dispersion function G(ω,k) times the am-plitude squared.More generally, for weakly nonlinear and slowly modu-lated waves propagating in one space dimension and in-cluding higher-order dispersion effects – not neglectingthe time and space derivatives ∂ta and ∂xa of the ampli-tude a(µx, µt) when taking derivatives, where µ ≪ 1 isa small modulation parameter – the averaged Lagrangiandensity is of the form:[22]
L = G(ω, k)a2+G2(ω, k)a4+1
2µ2(Gωω(∂Ta)
2 + 2Gωk(∂Ta)(∂Xa) +Gkk(∂Xa)2),
with the slow variables X = µx and T = µt.
5.6 References
5.6.1 Notes[1] Grimshaw (1984)
[2] Janssen (2004, pp. 16–24)
[3] Dewar (1970)
[4] Craik (1988, p. 17)
[5] Whitham (1974, pp. 395–397)
[6] Bretherton & Garrett (1968)
[7] Whitham (1974, p. 382)
[8] Whitham (1965)
[9] Simmons (1969)
[10] Willebrand (1975)
[11] Hayes (1973)
[12] Yuen & Lake (1975)
[13] Jimenez & Whitham (1976)
[14] Whitham (1974, pp. 390–397)
[15] Whitham (1974, pp. 522–523)
[16] Whitham (1974, p. 487)
[17] Whitham (1974, pp. 491–510)
40 CHAPTER 5. AVERAGED LAGRANGIAN
[18] Whitham (1974, p. 385)
[19] Whitham (1974, p. 498)
[20] Whitham (1974, §§16.6–16.13)
[21] Andrews & McIntyre (1978)
[22] Whitham (1974, pp. 522–526)
5.6.2 Publications by Whitham on themethod
An overview can be found in the book:
• Whitham, G.B. (1974), Linear and nonlinear waves,Wiley-Interscience, ISBN 0-471-94090-9
Some publications by Whitham on the method are:
• Whitham, G.B. (1965), “A general approachto linear and non-linear dispersive waves us-ing a Lagrangian”, Journal of Fluid Mechanics22 (2): 273–283, Bibcode:1965JFM....22..273W,doi:10.1017/S0022112065000745
• —— (1967a). “Non-linear dispersion of wa-ter waves”. Journal of Fluid Mechanics 27(2): 399–412. Bibcode:1967JFM....27..399W.doi:10.1017/S0022112067000424.
• —— (1967b), “Variational methods andapplications to water waves”, Proceedingsof the Royal Society of London. SeriesA. Mathematical and Physical Sciences 299(1456): 6–25, Bibcode:1967RSPSA.299....6W,doi:10.1098/rspa.1967.0119
• —— (1970), “Two-timing, variational princi-ples and waves”, Journal of Fluid Mechanics44 (2): 373–395, Bibcode:1970JFM....44..373W,doi:10.1017/S002211207000188X
• Jimenez, J.; Whitham, G.B. (1976), “An av-eraged Lagrangian method for dissipative wave-trains”, Proceedings of the Royal Society of Lon-don. A. Mathematical and Physical Sciences 349(1658): 277–287, Bibcode:1976RSPSA.349..277J,doi:10.1098/rspa.1976.0073
5.6.3 Further reading
• Andrews, D.G.; McIntyre, M.E. (1978),“On wave-action and its relatives” (PDF),Journal of Fluid Mechanics 89 (4):647–664, Bibcode:1978JFM....89..647A,doi:10.1017/S0022112078002785
• Bretherton, F.P.; Garrett, C.J.R. (1968), “Wave-trains in inhomogeneous moving media”, Proceed-ings of the Royal Society of London. Series A.Mathematical and Physical Sciences 302 (1471):529–554, Bibcode:1968RSPSA.302..529B,doi:10.1098/rspa.1968.0034
• Craik, A.D.D. (1988), Wave interactions andfluid flows, Cambridge University Press, ISBN9780521368292
• Dewar, R.L. (1970), “Interaction between hy-dromagnetic waves and a time‐dependent, in-homogeneous medium”, Physics of Fluids 13(11): 2710–2720, Bibcode:1970PhFl...13.2710D,doi:10.1063/1.1692854, ISSN 0031-9171
• Grimshaw, R. (1984), “Wave action and wave–mean flow interaction, with application to strati-fied shear flows”, Annual Review of Fluid Mechan-ics 16: 11–44, Bibcode:1984AnRFM..16...11G,doi:10.1146/annurev.fl.16.010184.000303
• Hayes, W.D. (1970), “Conservation of ac-tion and modal wave action”, Proceedingsof the Royal Society of London. A. Math-ematical and Physical Sciences 320 (1541):187–208, Bibcode:1970RSPSA.320..187H,doi:10.1098/rspa.1970.0205
• Hayes, W.D. (1973), “Group velocity and non-linear dispersive wave propagation”, Proceedingsof the Royal Society of London. A. Math-ematical and Physical Sciences 332 (1589):199–221, Bibcode:1973RSPSA.332..199H,doi:10.1098/rspa.1973.0021
• Holm, D.D. (2002), “Lagrangian averages,averaged Lagrangians, and the mean effectsof fluctuations in fluid dynamics”, Chaos 12(2): 518–530, Bibcode:2002Chaos..12..518H,doi:10.1063/1.1460941
• Janssen, P.A.E.M. (2004), The Interaction of OceanWaves and Wind, Cambridge University Press,ISBN 9780521465403
• Radder, A.C. (1999), “Hamiltonian dynamics ofwater waves”, in Liu, P.L.-F., Advances in Coastaland Ocean Engineering 4, World Scientific, pp. 21–59, ISBN 9789810233105
• Sedletsky, Y.V. (2012), “Addition of disper-sive terms to the method of averaged La-grangian”, Physics of Fluids 24 (6): 062105(15 pp.), Bibcode:2012PhFl...24f2105S,doi:10.1063/1.4729612
• Simmons, W.F. (1969), “A variational methodfor weak resonant wave interactions”, Pro-ceedings of the Royal Society of London. A.Mathematical and Physical Sciences 309 (1499):
5.6. REFERENCES 41
551–577, Bibcode:1969RSPSA.309..551S,doi:10.1098/rspa.1969.0056
• Willebrand, J. (1975), “Energy transport in anonlinear and inhomogeneous random grav-ity wave field”, Journal of Fluid Mechanics 70(1): 113–126, Bibcode:1975JFM....70..113W,doi:10.1017/S0022112075001929
• Yuen, H.C.; Lake, B.M. (1975), “Non-linear deep water waves: Theory andexperiment”, Physics of Fluids 18 (8):956–960, Bibcode:1975PhFl...18..956Y,doi:10.1063/1.861268
• Yuen, H.C.; Lake, B.M. (1980), “Insta-bilities of waves on deep water”, An-nual Review of Fluid Mechanics 12: 303–334, Bibcode:1980AnRFM..12..303Y,doi:10.1146/annurev.fl.12.010180.001511
Chapter 6
Canonical coordinates
In mathematics and classical mechanics, canonical co-ordinates are sets of coordinates which can be used todescribe a physical system at any given point in time (lo-cating the system within phase space). Canonical coordi-nates are used in the Hamiltonian formulation of classicalmechanics. A closely related concept also appears inquantum mechanics; see the Stone–von Neumann theo-rem and canonical commutation relations for details.As Hamiltonian mechanics is generalized by symplecticgeometry and canonical transformations are generalizedby contact transformations, so the 19th century definitionof canonical coordinates in classical mechanics may begeneralized to a more abstract 20th century definition ofcoordinates on the cotangent bundle of a manifold.
6.1 Definition, in classical mechan-ics
In classical mechanics, canonical coordinates are co-ordinates qi and pi in phase space that are used in theHamiltonian formalism. The canonical coordinates sat-isfy the fundamental Poisson bracket relations:
qi, qj = 0 pi, pj = 0 qi, pj = δij
A typical example of canonical coordinates is for qi to bethe usual Cartesian coordinates, and pi to be the compo-nents of momentum. Hence in general, the pi coordinatesare referred to as “conjugate momenta.”Canonical coordinates can be obtained from thegeneralized coordinates of the Lagrangian formalismby a Legendre transformation, or from another set ofcanonical coordinates by a canonical transformation.
6.2 Definition, on cotangent bun-dles
Canonical coordinates are defined as a special set ofcoordinates on the cotangent bundle of a manifold. Theyare usually written as a set of (qi, pj) or (xi, pj) with
the x 's or q 's denoting the coordinates on the underlyingmanifold and the p 's denoting the conjugate momen-tum, which are 1-forms in the cotangent bundle at pointq in the manifold.A common definition of canonical coordinates is any setof coordinates on the cotangent bundle that allow thecanonical one-form to be written in the form
∑i
pi dqi
up to a total differential. A change of coordinates thatpreserves this form is a canonical transformation; theseare a special case of a symplectomorphism, which are es-sentially a change of coordinates on a symplectic mani-fold.In the following exposition, we assume that the manifoldsare real manifolds, so that cotangent vectors acting on tan-gent vectors produce real numbers.
6.3 Formal development
Given amanifoldQ, a vector fieldX onQ (or equivalently,a section of the tangent bundle TQ) can be thought of asa function acting on the cotangent bundle, by the dualitybetween the tangent and cotangent spaces. That is, definea function
PX : T ∗Q → R
such that
PX(q, p) = p(Xq)
holds for all cotangent vectors p in T ∗q Q . Here, Xq is
a vector in TqQ , the tangent space to the manifold Qat point q. The function PX is called the momentumfunction corresponding to X.In local coordinates, the vector field X at point q may bewritten as
42
6.7. EXTERNAL LINKS 43
Xq =∑i
Xi(q)∂
∂qi
where the ∂/∂qi are the coordinate frame on TQ. Theconjugate momentum then has the expression
PX(q, p) =∑i
Xi(q) pi
where the pi are defined as the momentum functions cor-responding to the vectors ∂/∂qi :
pi = P∂/∂qi
The qi together with the pj together form a coordinatesystem on the cotangent bundle T ∗Q ; these coordinatesare called the canonical coordinates.
6.4 Generalized coordinates
In Lagrangian mechanics, a different set of coordinatesare used, called the generalized coordinates. These arecommonly denoted as (qi, qi) with qi called the gener-alized position and qi the generalized velocity. Whena Hamiltonian is defined on the cotangent bundle, thenthe generalized coordinates are related to the canonicalcoordinates by means of the Hamilton–Jacobi equations.
6.5 See also• Linear discriminant analysis
• symplectic manifold
• symplectic vector field
• symplectomorphism
• Kinetic momentum
6.6 References• Goldstein, Herbert; Poole, Charles P., Jr.; Safko,John L. (2002). Classical Mechanics (3rd ed.). SanFrancisco: Addison Wesley. pp. 347–349. ISBN0-201-65702-3.
6.7 External links
Chapter 7
Classical field theory
A classical field theory is a physical theory that de-scribes the study of how one or more physical fields inter-act with matter. The word 'classical' is used in contrast tothose field theories that incorporate quantum mechanics(quantum field theories).A physical field can be thought of as the assignment of aphysical quantity at each point of space and time. Forexample, in a weather forecast, the wind velocity dur-ing a day over a country is described by assigning a vec-tor to each point in space. Each vector represents thedirection of the movement of air at that point. As theday progresses, the directions in which the vectors pointchange as the directions of the wind change. From themathematical viewpoint, classical fields are described bysections of fiber bundles (covariant classical field the-ory). The term 'classical field theory' is commonly re-served for describing those physical theories that describeelectromagnetism and gravitation, two of the fundamentalforces of nature.Descriptions of physical fields were given before the ad-vent of relativity theory and then revised in light of thistheory. Consequently, classical field theories are usuallycategorised as non-relativistic and relativistic.
7.1 Non-relativistic field theories
Some of the simplest physical fields are vector force fields.Historically, the first time fields were taken seriously waswith Faraday’s lines of force when describing the electricfield. The gravitational field was then similarly described.
7.1.1 Newtonian gravitation
A classical field theory describing gravity is Newtoniangravitation, which describes the gravitational force as amutual interaction between two masses.Any massive bodyM has a gravitational field g which de-scribes its influence on other massive bodies. The gravi-tational field ofM at a point r in space is found by deter-mining the force F that M exerts on a small test mass mlocated at r, and then dividing by m:[1]
g(r) = F(r)m
.
Stipulating thatm is much smaller thanM ensures that thepresence of m has a negligible influence on the behaviorof M.According to Newton’s law of universal gravitation, F(r)is given by[1]
F(r) = −GMm
r2r,
where r is a unit vector pointing along the line fromM tom. Therefore, the gravitational field ofM is[1]
g(r) = F(r)m
= −GM
r2r.
The experimental observation that inertial mass and grav-itational mass are equal to unprecedented levels of accu-racy leads to the identification of the gravitational fieldstrength as identical to the acceleration experienced bya particle. This is the starting point of the equivalenceprinciple, which leads to general relativity.Because the gravitational force F is conservative, thegravitational field g can be written in terms of the gradientof a gravitational potential Φ(r):
g(r) = −∇Φ(r).
7.1.2 Electromagnetism
Electrostatics
Main article: Electrostatics
A charged test particle with charge q experiences a forceF based solely on its charge. We can similarly de-scribe the electric field E so that F = qE. Using this and
44
7.2. RELATIVISTIC FIELD THEORY 45
Coulomb’s law tells us that the electric field due to a singlecharged particle as
E =1
4πϵ0
q
r2r.
The electric field is conservative, and hence can be de-scribed by a scalar potential, V(r):
E(r) = −∇V (r).
Magnetostatics
Main article: Magnetostatics
A steady current I flowing along a path ℓ will exert a forceon nearby charged particles that is quantitatively differentfrom the electric field force described above. The forceexerted by I on a nearby charge q with velocity v is
F(r) = qv× B(r),
where B(r) is the magnetic field, which is determinedfrom I by the Biot–Savart law:
B(r) = µ0I
4π
∫dℓ× dr
r2.
The magnetic field is not conservative in general, andhence cannot usually be written in terms of a scalar po-tential. However, it can be written in terms of a vectorpotential, A(r):
B(r) = ∇× A(r)
Electrodynamics
Main article: Electrodynamics
In general, in the presence of both a charge density ρ(r, t)and current density J(r, t), there will be both an electricand a magnetic field, and both will vary in time. They aredetermined by Maxwell’s equations, a set of differentialequations which directly relate E and B to ρ and J.[note 1][2]
Alternatively, one can describe the system in terms of itsscalar and vector potentials V and A. A set of integralequations known as retarded potentials allow one to calcu-late V and A from ρ and J,[note 2] and from there the elec-tric and magnetic fields are determined via the relations[3]
E = −∇V − ∂A∂t
B = ∇× A.
7.1.3 Hydrodynamics
Main article: Hydrodynamics
Fluid dynamics has fields of pressure, density, and flowrate that are connected by conservation laws for energyand momentum. The mass continuity equation and New-ton’s laws connect the density, pressure, and velocityfields:
u = F− ∇p
ρ
ρ+∇ · (ρu) = 0
Here vector field is the velocity field.
7.2 Relativistic field theory
Main article: Covariant classical field theory
Modern formulations of classical field theories generallyrequire Lorentz covariance as this is now recognised as afundamental aspect of nature. A field theory tends to beexpressed mathematically by using Lagrangians. This is afunction that, when subjected to an action principle, givesrise to the field equations and a conservation law for thetheory.We use units where c , the speed of light in vacuum,equals 1, throughout.[note 3]
7.2.1 Lagrangian dynamics
Main article: Lagrangian (field theory)
Given a field tensor ϕ , a scalar called the Lagrangian den-sity L(ϕ, ∂ϕ, ∂∂ϕ, ..., x) can be constructed from ϕ andits derivatives.From this density, the functional action can be con-structed by integrating over spacetime
S =
∫Ld4x.
Therefore the Lagrangian itself is equal to the integral ofthe Lagrangian Density over all space.
46 CHAPTER 7. CLASSICAL FIELD THEORY
Then by enforcing the action principle, the Euler–Lagrange equations are obtained
δSδϕ
=∂L∂ϕ
−∂µ
(∂L
∂(∂µϕ)
)+. . .+(−1)m∂µ1∂µ2 . . .∂µm−1∂µm
(∂L
∂(∂µ1∂µ2 ...∂µm−1∂µmϕ)
)= 0.
7.3 Relativistic fields
Two of the most well-known Lorentz-covariant classicalfield theories are now described.
7.3.1 Electromagnetism
Main articles: Electromagnetic field andElectromagnetism
Historically, the first (classical) field theories were thosedescribing the electric and magnetic fields (separately).After numerous experiments, it was found that thesetwo fields were related, or, in fact, two aspects of thesame field: the electromagnetic field. Maxwell's theoryof electromagnetism describes the interaction of chargedmatter with the electromagnetic field. The first formula-tion of this field theory used vector fields to describe theelectric and magnetic fields. With the advent of specialrelativity, a more complete formulation using tensor fieldswas found. Instead of using two vector fields describingthe electric andmagnetic fields, a tensor field representingthese two fields together is used.
We have the electromagnetic potential, Aa =(−ϕ, A
), and the electromagnetic four-current ja =
(−ρ, j
).
The electromagnetic field at any point in spacetime is de-scribed by the antisymmetric (0,2)-rank electromagneticfield tensor
Fab = ∂aAb − ∂bAa.
The Lagrangian
To obtain the dynamics for this field, we try and con-struct a scalar from the field. In the vacuum, we haveL = −1
4µ0F abFab. We can use gauge field theory to get
the interaction term, and this gives us
L =−1
4µ0F abFab + jaAa.
The equations
This, coupled with the Euler–Lagrange equations, givesus the desired result, since the E–L equations say that
∂b
(∂L
∂ (∂bAa)
)=
∂L∂Aa
.
It is easy to see that ∂L/∂Aa = µ0ja . The left hand side
is trickier. Being careful with factors of F ab , however,the calculation gives ∂L/∂(∂bAa) = F ab . Together,then, the equations of motion are:
∂bFab = µ0j
a.
This gives us a vector equation, which are Maxwell’sequations in vacuum. The other two are obtained fromthe fact that F is the 4-curl of A :
6F[ab,c] = Fab,c + Fca,b + Fbc,a = 0.
where the comma indicates a partial derivative.
7.3.2 Gravitation
Main articles: Gravitation and General Relativity
After Newtonian gravitation was found to be incon-sistent with special relativity, Albert Einstein formu-lated a new theory of gravitation called general relativ-ity. This treats gravitation as a geometric phenomenon('curved spacetime') caused by masses and represents thegravitational field mathematically by a tensor field calledthe metric tensor. The Einstein field equations describehow this curvature is produced. The field equations maybe derived by using the Einstein–Hilbert action. Varyingthe Lagrangian
L = R√−g , (where R = Rabg
ab is the Ricciscalar written in terms of the Ricci tensor Rab andthe metric tensor gab )
will yield the vacuum field equations:
Gab = 0
(where Gab = Rab − R2 gab is the Einstein tensor).
7.4 See also• Classical unified field theories
• Covariant Hamiltonian field theory
• Variational methods in general relativity
• Higgs field (classical)
• Hamiltonian field theory
7.7. EXTERNAL LINKS 47
7.5 Notes[1] Where ρ is electric charge density (charge per unit vol-
ume) and J is current density (current flow per unit area)
[2] This is contingent on the correct choice of gauge. V andAare not completely determined by ρ and J; rather, they areonly determined up to some scalar function f(r, t) knownas the gauge. The retarded potential formalism requiresone to choose the Lorentz gauge.
[3] This is equivalent to choosing units of distance and timeas light-seconds and seconds or light-years and years.Choosing c = 1 allows us to simplify the equations. Forinstance: E = mc2 reduces to E = m (since c2 = 1,without keeping track of units). This reduces complexityof the expressions while keeping focus on the underlyingprinciples. This “trick” can't be used for actual numericalcalculations.
7.6 References[1] Kleppner, David; Kolenkow, Robert. An Introduction to
Mechanics. p. 85.
[2] Griffiths, David. Introduction to Electrodynamics (3rded.). p. 326.
[3] Wangsness, Roald. Electromagnetic Fields (2nd ed.). p.469.
• Truesdell, C.; Toupin, R.A. (1960). “The ClassicalField Theories”. In Flügge, Siegfried. Principles ofClassical Mechanics and Field Theory/Prinzipien derKlassischen Mechanik und Feldtheorie. Handbuchder Physik (Encyclopedia of Physics). III/1. Berlin–Heidelberg–New York: Springer-Verlag. pp. 226–793. Zbl 0118.39702.
7.7 External links• Thidé, Bo. “Electromagnetic Field Theory” (PDF).Retrieved February 14, 2006.
• Carroll, Sean M. “Lecture Notes on Gen-eral Relativity”. arXiv:gr-qc/9712019.Bibcode:1997gr.qc....12019C.
• Binney, James J. “LectureNotes onClassical Fields”(PDF). Retrieved April 30, 2007.
• Sardanashvily, G. (November 2008). “AdvancedClassical Field Theory”. International Journalof Geometric Methods in Modern Physics (WorldScientific) 5 (7): 1163. arXiv:0811.0331.Bibcode:2008IJGMM..05.1163S.doi:10.1142/S0219887808003247. ISBN 978-981-283-895-7.
Chapter 8
Covariant classical field theory
In mathematical physics, covariant classical field the-ory represents classical fields by sections of fiber bundles,and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well knownthat jet bundles and the variational bicomplex are thecorrect domain for such a description. The Hamiltonianvariant of covariant classical field theory is the covariantHamiltonian field theory where momenta correspond toderivatives of field variables with respect to all world co-ordinates. Non-autonomous mechanics is formulated ascovariant classical field theory on fiber bundles over thetime axis ℝ.
8.1 See also
• Classical field theory
• Exterior algebra
• Lagrangian system
• Variational bicomplex
• Quantum field theory
• Non-autonomous mechanics
• Higgs field (classical)
8.2 References
• Saunders, D.J., “The Geometry of Jet Bundles”,Cambridge University Press, 1989, ISBN 0-521-36948-7
• Bocharov, A.V. [et al.] “Symmetries and conserva-tion laws for differential equations of mathematicalphysics”, Amer. Math. Soc., Providence, RI, 1999,ISBN 0-8218-0958-X
• De Leon, M., Rodrigues, P.R., “Generalized Classi-cal Mechanics and Field Theory”, Elsevier SciencePublishing, 1985, ISBN 0-444-87753-3
• Griffiths, P.A., “Exterior Differential Systems andthe Calculus of Variations”, Boston: Birkhauser,1983, ISBN 3-7643-3103-8
• Gotay, M.J., Isenberg, J., Marsden, J.E., Mont-gomery R., Momentum Maps and Classical FieldsPart I: Covariant Field Theory, November 2003
• Echeverria-Enriquez, A., Munoz-Lecanda, M.C.,Roman-Roy,M., Geometry of Lagrangian First-order Classical Field Theories, May 1995
• Giachetta, G., Mangiarotti, L., Sardanashvily, G.,“Advanced Classical Field Theory”, World Sci-entific, 2009, ISBN 978-981-283-895-7 (arXiv:0811.0331v2)
48
Chapter 9
D'Alembert’s principle
Jean d'Alembert (1717—1783)
D'Alembert’s principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamen-tal classical laws of motion. It is named after its dis-coverer, the French physicist and mathematician Jean leRond d'Alembert. It is the dynamic analogue to the prin-ciple of virtual work for applied forces in a static sys-tem and in fact is more general than Hamilton’s princi-ple, avoiding restriction to holonomic systems.[1] A holo-nomic constraint depends only on the coordinates andtime. It does not depend on the velocities. If the negativeterms in accelerations are recognized as inertial forces,the statement of d'Alembert’s principle becomes The to-tal virtual work of the impressed forces plus the inertialforces vanishes for reversible displacements.[2] The prin-ciple does not apply for irreversible displacements, suchas sliding friction, and more general specification of theirreversibility is required.[3]
The principle states that the sum of the differences be-tween the forces acting on a system of mass particlesand the time derivatives of the momenta of the system
itself along any virtual displacement consistent with theconstraints of the system, is zero. Thus, in symbolsd'Alembert’s principle is written as following,
∑i
(Fi −miai) · δri = 0,
where
This above equation is often called d'Alembert’s prin-ciple, but it was first written in this variational formby Joseph Louis Lagrange.[4] D'Alembert’s contributionwas to demonstrate that in the totality of a dynamic sys-tem the forces of constraint vanish. That is to say thatthe generalized forces Qj need not include constraintforces. It is equivalent to the somewhat more cumber-some Gauss’s principle of least constraint.
9.1 General case with changingmasses
The general statement of d'Alembert’s principle mentions“the time derivatives of the momenta of the system”. Themomentum of the i-th mass is the product of its mass andvelocity:
pi = mivi
and its time derivative is
pi = mivi +mivi
In many applications, the masses are constant and thisequation reduces to
pi = mivi = miai
49
50 CHAPTER 9. D'ALEMBERT’S PRINCIPLE
which appears in the formula given above. However,some applications involve changing masses (for exam-ple, chains being rolled up or being unrolled) and in thosecases both terms mivi andmivi have to remain present,giving
∑i
(Fi −miai − mivi) · δri = 0.
9.2 Derivation for special cases
To date, nobody has shown that D'Alembert’s principleis equivalent to Newton’s Second Law. This is true onlyfor some very special cases e.g. rigid body constraints.However, an approximate solution to this problem doesexist.[5]
Consider Newton’s law for a system of particles, i. Thetotal force on each particle is[6]
F(T )i = miai,
where
Moving the inertial forces to the left gives an expressionthat can be considered to represent quasi-static equilib-rium, but which is really just a small algebraic manipula-tion of Newton’s law:[6]
F(T )i −miai = 0.
Considering the virtual work, δW , done by the total andinertial forces together through an arbitrary virtual dis-placement, δri , of the system leads to a zero identity,since the forces involved sum to zero for each particle.[6]
δW =∑i
F(T )i · δri −
∑i
miai · δri = 0
The original vector equation could be recovered by rec-ognizing that the work expression must hold for arbitrarydisplacements. Separating the total forces into appliedforces, Fi , and constraint forces, Ci , yields[6]
δW =∑i
Fi · δri +∑i
Ci · δri −∑i
miai · δri = 0.
If arbitrary virtual displacements are assumed to be indirections that are orthogonal to the constraint forces(which is not usually the case, so this derivation worksonly for special cases), the constraint forces do no
work. Such displacements are said to be consistentwith the constraints.[7] This leads to the formulation ofd'Alembert’s principle, which states that the difference ofapplied forces and inertial forces for a dynamic systemdoes no virtual work:.[6]
δW =∑i
(Fi −miai) · δri = 0.
There is also a corresponding principle for static systemscalled the principle of virtual work for applied forces.
9.3 D'Alembert’s principle of iner-tial forces
D'Alembert showed that one can transform an accelerat-ing rigid body into an equivalent static system by addingthe so-called "inertial force" and "inertial torque" or mo-ment. The inertial force must act through the center ofmass and the inertial torque can act anywhere. The sys-tem can then be analyzed exactly as a static system sub-jected to this “inertial force and moment” and the ex-ternal forces. The advantage is that, in the equivalentstatic system one can take moments about any point (notjust the center of mass). This often leads to simpler cal-culations because any force (in turn) can be eliminatedfrom the moment equations by choosing the appropriatepoint about which to apply the moment equation (sum ofmoments = zero). Even in the course of Fundamentalsof Dynamics and Kinematics of machines, this principlehelps in analyzing the forces that act on a link of a mech-anism when it is in motion. In textbooks of engineer-ing dynamics this is sometimes referred to as d'Alembert’sprinciple.
9.3.1 Example for 1D motion of a rigidbody
Free body diagram of a wire pulling on a mass with weight W,showing the d’Alembert inertia “force” ma.
To illustrate the concept of d'Alembert’s principle, let’s usea simple model with a weight W , suspended from a wire.
9.4. REFERENCES[8] 51
The weight is subjected to a gravitational force,W = mg, and a tension force T in the wire. The mass acceleratesupward with an acceleration a . Newton’s Second Lawbecomes T − W = ma or T = W + ma . As anobserver with feet planted firmly on the ground, we seethat the force T accelerates the weight, W , but, if weare moving with the wire we don’t see the acceleration,we feel it. The tension in the wire seems to counteract anacceleration “force”ma or (W/g)a .
Free body diagram depicting an inertia moment and an inertiaforce on a rigid body in free fall with an angular velocity.
9.3.2 Example for plane 2D motion of arigid body
For a planar rigid body, moving in the plane of the body(the x–y plane), and subjected to forces and torques caus-ing rotation only in this plane, the inertial force is
Fi = −mrc
where rc is the position vector of the centre of mass ofthe body, and m is the mass of the body. The inertialtorque (or moment) is
Ti = −Iθ
where I is the moment of inertia of the body. If, in ad-dition to the external forces and torques acting on thebody, the inertia force acting through the center of massis added and the inertial torque is added (acting aroundthe centre of mass is as good as anywhere) the system isequivalent to one in static equilibrium. Thus the equa-tions of static equilibrium
∑Fx = 0,∑Fy = 0,∑T = 0
hold. The important thing is that∑
T is the sum oftorques (or moments, including the inertial moment andthe moment of the inertial force) taken about any point.The direct application of Newton’s laws requires that theangular acceleration equation be applied only about thecenter of mass.
9.3.3 Dynamic equilibrium
D'Alembert’s form of the principle of virtual work statesthat a system of rigid bodies is in dynamic equilibriumwhen the virtual work of the sum of the applied forcesand the inertial forces is zero for any virtual displacementof the system. Thus, dynamic equilibrium of a systemof n rigid bodies with m generalized coordinates requiresthat is to be
δW = (Q1 +Q∗1)δq1 + . . .+ (Qm +Q∗
m)δqm = 0,
for any set of virtual displacements δq . This conditionyields m equations,
Qj +Q∗j = 0, j = 1, . . . ,m,
which can also be written as
d
dt
∂T
∂qj− ∂T
∂qj= Qj , j = 1, . . . ,m.
The result is a set of m equations of motion that definethe dynamics of the rigid body system.
9.4 References[8]
[1] Lanczos, Cornelius (1970). The Variational Principles ofMechanics (4th ed.). New York: Dover Publications Inc.p. 92. ISBN 0-486-65067-7.
[2] Cornelius Lanczos (1970). p. 90. ISBN 0-486-65067-7.
[3] Udwadia, F. E.; Kalaba, R. E. (2002). “On the Founda-tions of Analytical Dynamics” (PDF). Intl. Journ. Nonlin-ear Mechanics 37 (6): 1079–1090. doi:10.1016/S0020-7462(01)00033-6.
[4] Arnold Sommerfeld (1956),Mechanics: Lectures on The-oretical Physics, Vol 1, p. 53
[5] Rebhan, Eckhard (2006). “Exkurs 5.1: Ableitung desd'Alembert Prinzips”. Mechanik. Theoretische Physik.Heidelberg, Germany: Spektrum Akademischer Verlag.ISBN 978-3-8274-1716-9.
[6] Torby, Bruce (1984). “Energy Methods”. Advanced Dy-namics for Engineers. HRW Series in Mechanical Engi-neering. United States of America: CBS College Publish-ing. ISBN 0-03-063366-4.
52 CHAPTER 9. D'ALEMBERT’S PRINCIPLE
[7] Jong, Ing-Chang (2005). “ImprovingMechanics of Mate-rials”. Teaching Students Work and Virtual Work Methodin Statics:A Guiding Strategy with Illustrative Examples(PDF). 2005 American Society for Engineering Educa-tion Annual Conference & Exposition. Retrieved June 24,2014.
[8] Weisshaar, Terry (2009). Aerospace Structures - an Intro-duction to Fundamental Problems. Purdue University. pp.50,58.
Chapter 10
Fiber derivative
In the context of Lagrangian Mechanics the fiber deriva-tive is used to convert between the Lagrangian andHamiltonian forms. In particular, if Q is the configura-tion manifold then the Lagrangian L is defined on thetangent bundle TQ and the Hamiltonian is defined onthe cotangent bundle T ∗Q—the fiber derivative is a mapFL : TQ → T ∗Q such that
FL(v) · w =d
ds|s=0L(v + sw)
where v and w are vectors from the same tangent space.When restricted to a particular point, the fiber derivativeis a Legendre transformation.
53
Chapter 11
FLEXPART
The FLEXible PARTicle dispersion model (FLEX-PART) is a Lagrangian Particle Dispersion Model usedto simulate air parcel trajectories. It can be run in eitherforward or backward mode. The forward mode is typi-cally used to determine the downwind concentration ormixing ratio of pollutants. The backward mode can beused to estimate footprint areas, to determine the originof observed emissions.
11.1 History
FLEXPART has inherited large portions of its codefrom its predecessor, FLEXTRA (FLEXible TRAjectorymodel). The first version of FLEXPART was released in1998 and was considered free software. Since the releaseof version 8.2 in 2010, the code is distributed under theGNU General Public License (GPL) Version 3.[1] Due toa growing user base the main developers decided to pro-vide an online platform for developers and modellers,which was launched together with the release of version9.0 in June 2012.In addition to main FLEXPART code, several brancheshave been developed for use with mesoscale meteorolog-ical models. In particular, FLEXPART-WRF was cre-ated to work with the open source WRF model. The firstversion of FLEXPART-WRF was presented in 2006 byJerome D. Fast and Richard C. Easter.[2] The model waslater renamed the “PNNL Integrated Lagrangian Trans-port” (PILT) model since the code base started to de-viate extensively from the main FLEXPART branch.[3]In 2007, a new technical report was presented werethe model was once again referred to as FLEXPART-WRF.[4] At this time, there were still a number of impor-tant features missing in FLEXPART-WRF (which wereavailable in FLEXPART). A number of research groupsstarted developing FLEXPART-WRF on their own, in2011, there were three separate projects on GitHub, eachwith a partial goal to implement a scheme for wet depo-sition.In 2013, a major update of FLEXPART-WRF was re-leased with support from the FLEXPART developers, therelease included a working wet deposition scheme as wellas new run-time options for wind fields and turbulence.[5]
The code was also parallelized with compile-time optionsfor OPENMP and MPI added to the previous default se-rial mode. Furthermore, an option to use the NetCDFstandard format for output was added.
11.2 References[1] The Lagrangian particle dispersion model FLEXPART
version 8.2 Retrieved 24 Mar 2015
[2] A Lagrangian particle dispersion model compatible withWRF (pdf) Retrieved 24 Mar 2015
[3] Development of a Lagrangian Particle Dispersion ModelCompatible with the Weather, Research and Forecasting(WRF) Model-Phase 2, Oct 2006 by Jerome D. Fast andRichard C. Easter, Pacific Northwest National Laboratory
[4] Development of a Lagrangian Particle Dispersion ModelCompatible with the Weather Research and Forecasting(WRF) Model – Phase 3, Jan 2007 by Jerome D. Fast andRichard C. Easter, Pacific Northwest National Laboratory
[5] The Lagrangian particle dispersion model FLEXPART-WRF version 3.1
54
Chapter 12
Generalized forces
Generalized forces find use in Lagrangian mechanics,where they play a role conjugate to generalized coordi-nates. They are obtained from the applied forces, Fᵢ,i=1,..., n, acting on a system that has its configurationdefined in terms of generalized coordinates. In the for-mulation of virtual work, each generalized force is thecoefficient of the variation of a generalized coordinate.
12.1 Virtual work
Generalized forces can be obtained from the computationof the virtual work, δW, of the applied forces.[1]:265
The virtual work of the forces, Fᵢ, acting on the particlesPᵢ, i=1,..., n, is given by
δW =n∑
i=1
Fi · δri
where δrᵢ is the virtual displacement of the particle Pᵢ.
12.1.1 Generalized coordinates
Let the position vectors of each of the particles, rᵢ, bea function of the generalized coordinates, q , j=1,...,m.Then the virtual displacements δrᵢ are given by
δri =m∑j=1
∂ri∂qj
δqj , i = 1, . . . , n,
where δq is the virtual displacement of the generalizedcoordinate q .The virtual work for the system of particles becomes
δW = F1 ·m∑j=1
∂r1∂qj
δqj + . . .+ Fn ·m∑j=1
∂rn∂qj
δqj .
Collect the coefficients of δq so that
δW =n∑
i=1
Fi ·∂ri∂q1
δq1 + . . .+n∑
i=1
Fi ·∂ri∂qm
δqm.
12.1.2 Generalized forces
The virtual work of a system of particles can be writtenin the form
δW = Q1δq1 + . . .+Qmδqm,
where
Qj =n∑
i=1
Fi ·∂ri∂qj
, j = 1, . . . ,m,
are called the generalized forces associated with the gen-eralized coordinates q , j=1,...,m.
12.1.3 Velocity formulation
In the application of the principle of virtual work it is of-ten convenient to obtain virtual displacements from thevelocities of the system. For the n particle system, letthe velocity of each particle Pᵢ be Vᵢ, then the virtual dis-placement δrᵢ can also be written in the form[2]
δri =m∑j=1
∂Vi
∂qjδqj , i = 1, . . . , n.
This means that the generalized force, Q , can also be de-termined as
Qj =
n∑i=1
Fi ·∂Vi
∂qj, j = 1, . . . ,m.
12.2 D'Alembert’s principle
D'Alembert formulated the dynamics of a particle as theequilibrium of the applied forces with an inertia force(apparent force), called D'Alembert’s principle. The in-ertia force of a particle, Pᵢ, of mass mᵢ is
55
56 CHAPTER 12. GENERALIZED FORCES
F∗i = −miAi, i = 1, . . . , n,
where Aᵢ is the acceleration of the particle.If the configuration of the particle system depends on thegeneralized coordinates q , j=1,...,m, then the generalizedinertia force is given by
Q∗j =
n∑i=1
F∗i ·
∂Vi
∂qj, j = 1, . . . ,m.
D'Alembert’s form of the principle of virtual work yields
δW = (Q1 +Q∗1)δq1 + . . .+ (Qm +Q∗
m)δqm.
12.3 References[1] Torby, Bruce (1984). “Energy Methods”. Advanced Dy-
namics for Engineers. HRW Series in Mechanical Engi-neering. United States of America: CBS College Publish-ing. ISBN 0-03-063366-4.
[2] T. R. Kane and D. A. Levinson, Dynamics, Theory andApplications, McGraw-Hill, NY, 2005.
12.4 See also• Lagrangian mechanics
• Generalized coordinates
• Degrees of freedom (physics and chemistry)
• Virtual work
Chapter 13
Geometric mechanics
Geometric mechanics is a branch of mathematics apply-ing particular geometric methods to many areas of me-chanics, from mechanics of particles and rigid bodies tofluid mechanics to control theory.Geometric mechanics applies principally to systems forwhich the configuration space is a Lie group, or a group ofdiffeomorphisms, or more generally where some aspectof the configuration space has this group structure. Forexample, the configuration space of a rigid body such asa satellite is the group of Euclidean motions (translationsand rotations in space), while the configuration space fora liquid crystal is the group of diffeomorphisms coupledwith an internal state (gauge symmetry or order parame-ter).
13.1 Momentum map and reduc-tion
One of the principal ideas of Geometric Mechanics is re-duction, which goes back to Jacobi’s elimination of thenode in the 3-body problem, but in its modern form is dueto K. Meyer (1973) and independently J.E. Marsden andA. Weinstein (1974), both inspired by the work of Smale(1970). Symmetry of a Hamiltonian or Lagrangian sys-tem gives rise to conserved quantities, by Noether’s the-orem, and these conserved quantities are the componentsof the momentum map J. If P is the phase space andG the symmetry group, the momentum map is a mapJ : P → g∗ , and the reduced spaces are quotients of thelevel sets of J by the subgroup of G preserving the levelset in question: for µ ∈ g∗ one definesPµ = J−1(µ)/Gµ
, and this reduced space is a symplectic manifold if µ isa regular value of J.
13.2 Variational principles• Euler-Lagrange
• D'Alembert
• Maupertuis
• Euler-Poincaré
• Vakonomic
•
13.3 Geometric integrators
One of the important developments arising from the geo-metric approach to mechanics is the incorporation of thegeometry into numerical methods. In particular symplec-tic and variational integrators are proving particularly ac-curate for long-term integration of Hamiltonian and La-grangian systems.
13.4 History
As amodern subject, geometric mechanics has its roots infour works written in the 1960s. These were by VladimirArnold (1966), Stephen Smale (1970) and Jean-MarieSouriau (1970), and the first edition of Abraham andMarsden’s Foundation of Mechanics (1967). Arnold’sfundamental work showed that Euler’s equations for thefree rigid body are the equations for geodesic flow onthe rotation group SO(3) and carried this geometric in-sight over to the dynamics of ideal fluids, where the rota-tion group is replaced by the group of volume preservingdiffeomorphisms. Smale’s paper on Topology and Me-chanics investigates the conserved quantities arising fromNoether’s theorem when a Lie group of symmetries actson a mechanical system, and defines what is now calledthe momentum map (which Smale calls angular momen-tum), and he raises questions about the topology of theenergy-momentum level surfaces and the effect on thedynamics. In his book, Souriau also considers the con-served quantities arising from the action of a group ofsymmetries, but he concentrates more on the geometricstructures involved (for example the equivariance prop-erties of this momentum for a wide class of symmetries),and less on questions of dynamics.These ideas, and particularly those of Smale were centralin the second edition of Foundations of Mechanics (Abra-ham and Marsden, 1978).
57
58 CHAPTER 13. GEOMETRIC MECHANICS
13.5 Applications
• Computer graphics
• Control theory — see Bloch (2003)
• Liquid Crystals — see Gay-Balmaz, Ratiu, Tronci(2013)
• Magnetohydrodynamics
• Molecular oscillations
• Nonholonomic constraints — see Bloch (2003)
• Nonlinear stability
• Plasmas — see Holm, Marsden, Weinstein (1985)
• Superfluids
• Trajectory planning for space exploration
• Underwater vehicles
• Variational integrators
13.6 Notes
13.7 References
• Abraham, Ralph; Marsden, Jerrold E. (1978), Foun-dations of Mechanics (2nd ed.), Addison-Wesley
• Arnold, Vladimir (1966), “Sur la géométrie dif-férentielle des groupes de Lie de dimension infine etses applications a l'hydrodynamique des fluides par-faits”, Annales de l'Institut de Fourier 16: 319–361,doi:10.5802/aif.233
• Arnold, Vladimir (1978),Mathematical Methods forClassical Mechanics, Springer-Verlag
• Bloch, Anthony (2003). Nonholonomic Mechanicsand Control. Springer-Verlag.
• Gay-Balmaz, Francois; Ratiu, Tudor; Tronci, Ce-sare (2013). “Equivalent Theories of Liquid CrystalDynamics”. Arch. Ration. Mech. Anal. 210: 773–811. doi:10.1007/s00205-013-0673-1.
• Holm, Darryl D.; Marsden, Jerrold E.; Ratiu, TudorS.; Weinstein, Alan (1985). “Nonlinear stability offluid and plasma equilibria”. Physics Reports 123:1–116. doi:10.1016/0370-1573(85)90028-6.
• Libermann, Paulette; Marle, Charles-Michel(1987). Symplectic geometry and analytical me-chanics. Mathematics and its Applications 35.Dordrecht: D. Reidel. doi:10.1007/978-94-009-3807-6. ISBN 90-277-2438-5.
• Marsden, Jerrold; Weinstein, Alan (1974), “Re-duction of Symplectic Manifolds with Symme-try”, Reports on Mathematical Physics 5: 121–130,doi:10.1016/0034-4877(74)90021-4
• Marsden, Jerrold; Ratiu, Tudor S. (1999). Introduc-tion to mechanics and symmetry. Texts in AppliedMathematics (2 ed.). New York: Springer-Verlag.ISBN 0-387-98643-X.
• Meyer, Kenneth (1973), Symmetries and integrals inmechanics, New York: Academic Press, pp. 259–272
• Ortega, Juan-Pablo; Ratiu, Tudor S. (2004). Mo-mentum maps and Hamiltonian reduction. Progressin Mathematics 222. Birkhauser Boston. ISBN 0-8176-4307-9.
• Smale, Stephen (1970), “Topology and Mechan-ics I”, Inventiones Mathematicae 10: 305–331,doi:10.1007/bf01418778
• Souriau, Jean-Marie (1970), Structure des SystemesDynamiques, Dunod
Chapter 14
Gibbons–Hawking–York boundary term
In general relativity, the Gibbons–Hawking–Yorkboundary term is a term that needs to be added to theEinstein–Hilbert action when the underlying spacetimemanifold has a boundary.The Einstein–Hilbert action is the basis for the most ele-mentary variational principle from which the field equa-tions of general relativity can be defined. However, theuse of the Einstein–Hilbert action is appropriate onlywhen the underlying spacetime manifold M is closed,i.e., a manifold which is both compact and withoutboundary. In the event that the manifold has a boundary∂M , the action should be supplemented by a boundaryterm so that the variational principle is well-defined.The necessity of such a boundary term was first realisedby York and later refined in a minor way by Gibbons andHawking.For a manifold that is not closed, the appropriate actionis
SEH+SGHY =1
16π
∫M
d4x√gR+1
8π
∫∂M
d3x√hK,
where SEH is the Einstein–Hilbert action, SGHY is theGibbons–Hawking–York boundary term, hαβ is theinduced metric on the boundary andK is the trace of thesecond fundamental form. Varying the action with re-spect to the metric gαβ gives the Einstein equations; theaddition of the boundary term means that in performingthe variation, the geometry of the boundary encoded inthe inducedmetric hαβ is fixed. There remains ambiguityin the action up to an arbitrary functional of the inducedmetric hαβ .
14.1 References• York, J. W. (1972). “Role of conformalthree-geometry in the dynamics of gravi-tation”. Physical Review Letters 28 (16):1082. Bibcode:1972PhRvL..28.1082Y.doi:10.1103/PhysRevLett.28.1082.
• Gibbons, G. W.; Hawking, S. W. (1977).“Action integrals and partition functions in
quantum gravity”. Physical Review D 15(10): 2752. Bibcode:1977PhRvD..15.2752G.doi:10.1103/PhysRevD.15.2752.
59
Chapter 15
Hamilton’s principle
In physics, Hamilton’s principle is William RowanHamilton's formulation of the principle of stationary ac-tion (see that article for historical formulations). It statesthat the dynamics of a physical system is determined bya variational problem for a functional based on a sin-gle function, the Lagrangian, which contains all phys-ical information concerning the system and the forcesacting on it. The variational problem is equivalent toand allows for the derivation of the differential equationsof motion of the physical system. Although formulatedoriginally for classical mechanics, Hamilton’s principlealso applies to classical fields such as the electromagneticand gravitational fields, and has even been extended toquantum mechanics, quantum field theory and criticalitytheories.
As the system evolves, q traces a path through configuration space(only some are shown). The path taken by the system (red) hasa stationary action (δS = 0) under small changes in the configu-ration of the system (δq).[1]
15.1 Mathematical formulation
Hamilton’s principle states that the true evolution q(t) ofa system described by N generalized coordinates q = (q1,q2, ..., qN) between two specified states q1 = q(t1) andq2 = q(t2) at two specified times t1 and t2 is a stationarypoint (a point where the variation is zero), of the actionfunctional
S[q] def=∫ t2
t1
L(q(t), q(t), t) dt
where L(q, q, t) is the Lagrangian function for the sys-tem. In other words, any first-order perturbation of thetrue evolution results in (at most) second-order changesin S . The action S is a functional, i.e., something thattakes as its input a function and returns a single number,a scalar. In terms of functional analysis, Hamilton’s prin-ciple states that the true evolution of a physical system isa solution of the functional equation
15.1.1 Euler–Lagrange equations derivedfrom the action integral
Requiring that the true trajectory q(t) be a stationarypoint of the action functional S is equivalent to a setof differential equations for q(t) (the Euler–Lagrangeequations), which may be derived as follows.Let q(t) represent the true evolution of the system be-tween two specified states q1 = q(t1) and q2 = q(t2) attwo specified times t1 and t2, and let ε(t) be a small per-turbation that is zero at the endpoints of the trajectory
ε(t1) = ε(t2)def= 0
To first order in the perturbation ε(t), the change in theaction functional δS would be
δS =
∫ t2
t1
[L(q+ ε, q+ ε)− L(q, q)] dt =∫ t2
t1
(ε · ∂L
∂q + ε · ∂L∂q
)dt
where we have expanded the Lagrangian L to first orderin the perturbation ε(t).Applying integration by parts to the last term results in
δS =
[ε · ∂L
∂q
]t2t1
+
∫ t2
t1
(ε · ∂L
∂q − ε · d
dt
∂L
∂q
)dt
60
15.2. HAMILTON’S PRINCIPLE APPLIED TO DEFORMABLE BODIES 61
The boundary conditions ε(t1) = ε(t2)def= 0 causes the
first term to vanish
δS =
∫ t2
t1
ε ·(∂L
∂q − d
dt
∂L
∂q
)dt
Hamilton’s principle requires that this first-order changeδS is zero for all possible perturbations ε(t), i.e., the truepath is a stationary point of the action functional S (eithera minimum, maximum or saddle point). This require-ment can be satisfied if and only if
These equations are called the Euler–Lagrange equationsfor the variational problem.
15.1.2 Canonical momenta and constantsof motion
The conjugate momentum pk for a generalized coordi-nate qk is defined by the equation
pkdef=
∂L
∂qk
An important special case of the Euler–Lagrange equa-tion occurs when L does not contain a generalized coor-dinate qk explicitly,
∂L
∂qk= 0 ⇒ d
dt
∂L
∂qk= 0 ⇒ dpk
dt= 0 ,
that is, the conjugate momentum is a constant of the mo-tion.In such cases, the coordinate qk is called a cyclic coordi-nate. For example, if we use polar coordinates t, r, θ todescribe the planar motion of a particle, and if L does notdepend on θ, the conjugate momentum is the conservedangular momentum.
15.1.3 Example: Free particle in polar co-ordinates
Trivial examples help to appreciate the use of the actionprinciple via the Euler–Lagrange equations. A free par-ticle (mass m and velocity v) in Euclidean space movesin a straight line. Using the Euler–Lagrange equations,this can be shown in polar coordinates as follows. In theabsence of a potential, the Lagrangian is simply equal tothe kinetic energy
L =1
2mv2 =
1
2m(x2 + y2
)
in orthonormal (x,y) coordinates, where the dot repre-sents differentiation with respect to the curve parameter(usually the time, t). Therefore, upon application of theEuler–Lagrange equations,
d
dt
(∂L
∂x
)− ∂L
∂x= 0 ⇒ mx = 0
And likewise for y. Thus the Euler–Lagrange formulationcan be used to derive Newton’s laws.In polar coordinates (r, φ) the kinetic energy and hencethe Lagrangian becomes
L =1
2m(r2 + r2φ2
).
The radial r and φ components of the Euler–Lagrangeequations become, respectively
d
dt
(∂L
∂r
)− ∂L
∂r= 0 ⇒ r − rφ2 = 0
d
dt
(∂L
∂φ
)− ∂L
∂φ= 0 ⇒ φ+
2
rrφ = 0.
The solution of these two equations is given by
r =√
(at+ b)2 + c2
φ = tan−1
(at+ b
c
)+ d
for a set of constants a, b, c, d determined by initial con-ditions. Thus, indeed, the solution is a straight line givenin polar coordinates: a is the velocity, c is the distance ofthe closest approach to the origin, and d is the angle ofmotion.
15.2 Hamilton’s principle appliedto deformable bodies
Hamilton’s principle is an important variational principlein elastodynamics. As opposed to a system composed ofrigid bodies, deformable bodies have an infinite numberof degrees of freedom and occupy continuous regions ofspace; consequently, the state of the system is describedby using continuous functions of space and time. Theextended Hamilton Principle for such bodies is given by
∫ t2
t1
[δWe + δT − δU ] dt = 0
where T is the kinetic energy, U is the elastic energy,Weis the work done by external loads on the body, and t1, t2
62 CHAPTER 15. HAMILTON’S PRINCIPLE
the initial and final times. If the system is conservative,the work done by external forces may be derived from ascalar potential V. In this case,
δ
∫ t2
t1
[T − (U + V )] dt = 0.
This is called Hamilton’s principle and it is invariant un-der coordinate transformations.
15.3 Comparison with Mauper-tuis’ principle
Hamilton’s principle and Maupertuis’ principle are occa-sionally confused and both have been called (incorrectly)the principle of least action. They differ in three impor-tant ways:
• their definition of the action...
Maupertuis’ principle uses an integral over thegeneralized coordinates known as the abbrevi-ated action
S0def=
∫p · dq
where p = (p1, p2, ..., pN) are the conjugatemomenta defined above. By contrast, Hamil-ton’s principle uses S , the integral of theLagrangian over time.
• the solution that they determine...
Hamilton’s principle determines the trajectoryq(t) as a function of time, whereas Mauper-tuis’ principle determines only the shape of thetrajectory in the generalized coordinates. Forexample, Maupertuis’ principle determines theshape of the ellipse on which a particle movesunder the influence of an inverse-square cen-tral force such as gravity, but does not describeper se how the particle moves along that tra-jectory. (However, this time parameterizationmay be determined from the trajectory itself insubsequent calculations using the conservationof energy). By contrast, Hamilton’s principledirectly specifies the motion along the ellipseas a function of time.
• ...and the constraints on the variation.
Maupertuis’ principle requires that the twoendpoint states q1 and q2 be given and that en-ergy be conserved along every trajectory (same
energy for each trajectory). This forces theendpoint times to be varied as well. By con-trast, Hamilton’s principle does not require theconservation of energy, but does require thatthe endpoint times t1 and t2 be specified as wellas the endpoint states q1 and q2.
15.4 Action principle for fields
15.4.1 Classical field theory
Main article: Classical field theory
The action principle can be extended to obtain theequations ofmotion for fields, such as the electromagneticfield or gravity.The Einstein equation utilizes the Einstein–Hilbert actionas constrained by a variational principle.The path of a body in a gravitational field (i.e. free fall inspace time, a so-called geodesic) can be found using theaction principle.
15.4.2 Quantum mechanics and quantumfield theory
Main article: Quantum field theory
In quantum mechanics, the system does not follow a sin-gle path whose action is stationary, but the behavior ofthe system depends on all imaginable paths and the valueof their action. The action corresponding to the variouspaths is used to calculate the path integral, that gives theprobability amplitudes of the various outcomes.Although equivalent in classical mechanics with Newton’slaws, the action principle is better suited for general-izations and plays an important role in modern physics.Indeed, this principle is one of the great generalizationsin physical science. In particular, it is fully appreciatedand best understood within quantum mechanics. RichardFeynman's path integral formulation of quantum me-chanics is based on a stationary-action principle, usingpath integrals. Maxwell’s equations can be derived asconditions of stationary action.
15.5 See also
• Analytical mechanics
• Configuration space
• Hamilton–Jacobi equation
• Phase space
15.6. REFERENCES 63
• Geodesics as Hamiltonian flows
15.6 References[1] R. Penrose (2007). The Road to Reality. Vintage books.
p. 474. ISBN 0-679-77631-1.
• W.R. Hamilton, “On a General Method in Dynam-ics.”, Philosophical Transaction of the Royal Soci-ety Part II (1834) pp. 247–308; Part I (1835) pp.95–144. (From the collection Sir William RowanHamilton (1805–1865): Mathematical Papers editedby David R. Wilkins, School of Mathematics, TrinityCollege, Dublin 2, Ireland. (2000); also reviewed asOn a General Method in Dynamics)
• Goldstein H. (1980) Classical Mechanics, 2nd ed.,Addison Wesley, pp. 35–69.
• Landau LD and Lifshitz EM (1976)Mechanics, 3rd.ed., Pergamon Press. ISBN 0-08-021022-8 (hard-cover) and ISBN 0-08-029141-4 (softcover), pp. 2–4.
• Arnold VI. (1989)Mathematical Methods of Classi-cal Mechanics, 2nd ed., Springer Verlag, pp. 59–61.
• Cassel, Kevin W.: Variational Methods with Appli-cations in Science and Engineering, Cambridge Uni-versity Press, 2013.
Chapter 16
Inverse problem for Lagrangian mechanics
In mathematics, the inverse problem for Lagrangianmechanics is the problem of determining whether a givensystem of ordinary differential equations can arise as theEuler–Lagrange equations for some Lagrangian function.There has been a great deal of activity in the studyof this problem since the early 20th century. A no-table advance in this field was a 1941 paper by theAmerican mathematician Jesse Douglas, in which he pro-vided necessary and sufficient conditions for the problemto have a solution; these conditions are now known asthe Helmholtz conditions, after the German physicistHermann von Helmholtz.
16.1 Background and statement ofthe problem
The usual set-up of Lagrangian mechanics on n-dimensional Euclidean space Rn is as follows. Considera differentiable path u : [0, T] → Rn. The action of thepath u, denoted S(u), is given by
S(u) =
∫ T
0
L(t, u(t), u(t)) dt,
where L is a function of time, position and velocity knownas the Lagrangian. The principle of least action statesthat, given an initial state x0 and a final state x1 in Rn,the trajectory that the system determined by L will actu-ally follow must be a minimizer of the action functional Ssatisfying the boundary conditions u(0) = x0, u(T) = x1.Furthermore, the critical points (and hence minimizers)of S must satisfy the Euler–Lagrange equations for S:
ddt
∂L
∂ui− ∂L
∂ui= 0 for1 ≤ i ≤ n,
where the upper indices i denote the components of u =(u1, ..., un).In the classical case
T (u) =1
2m|u|2,
V : [0, T ]× Rn → R,
L(t, u, u) = T (u)− V (t, u),
the Euler–Lagrange equations are the second-order ordi-nary differential equations better known as Newton’s lawsof motion:
ui = −∂V (t, u)
∂uifor1 ≤ i ≤ n,
i.e. u = −∇uV (t, u).
The inverse problem of Lagrangian mechanics is asfollows: given a system of second-order ordinary differ-ential equations
ui = f i(uj , uj) for1 ≤ i, j ≤ n, (E)
that holds for times 0 ≤ t ≤ T, does there exist a La-grangian L : [0, T] × Rn × Rn → R for which these or-dinary differential equations (E) are the Euler–Lagrangeequations? In general, this problem is posed not on Eu-clidean space Rn, but on an n-dimensional manifold M,and the Lagrangian is a function L : [0, T] × TM → R,where TM denotes the tangent bundle of M.
16.2 Douglas’ theorem and theHelmholtz conditions
To simplify the notation, let
vi = ui
and define a collection of n2 functions Φji by
Φij =
1
2
ddt
∂f i
∂vj− ∂f i
∂uj− 1
4
∂f i
∂vk∂fk
∂vj.
Theorem. (Douglas 1941) There exists a LagrangianL : [0, T] × TM → R such that the equations (E) are
64
16.3. REFERENCES 65
its Euler–Lagrange equations if and only if there existsa non-singular symmetric matrix g with entries gij de-pending on both u and v satisfying the following threeHelmholtz conditions:
gΦ = (gΦ)⊤, (H1)
dgijdt +
1
2
∂fk
∂vigkj+
1
2
∂fk
∂vjgki = 0 for 1 ≤ i, j ≤ n, (H2)
∂gij∂vk
=∂gik∂vj
for 1 ≤ i, j, k ≤ n. (H3)
(The Einstein summation convention is in use for the re-peated indices.)
16.2.1 Applying Douglas’ theorem
At first glance, solving the Helmholtz equations (H1)–(H3) seems to be an extremely difficult task. Condition(H1) is the easiest to solve: it is always possible to finda g that satisfies (H1), and it alone will not imply thatthe Lagrangian is singular. Equation (H2) is a systemof ordinary differential equations: the usual theorems onthe existence and uniqueness of solutions to ordinary dif-ferential equations imply that it is, in principle, possibleto solve (H2). Integration does not yield additional con-stants but instead first integrals of the system (E), so thisstep becomes difficult in practice unless (E) has enoughexplicit first integrals. In certain well-behaved cases (e.g.the geodesic flow for the canonical connection on a Liegroup), this condition is satisfied.The final and most difficult step is to solve equation (H3),called the closure conditions since (H3) is the conditionthat the differential 1-form gi is a closed form for each i.The reason why this is so daunting is that (H3) constitutesa large system of coupled partial differential equations:for n degrees of freedom, (H3) constitutes a system of
2
(n+ 13
)partial differential equations in the 2n independent vari-ables that are the components gij of g, where
(nk
)denotes the binomial coefficient. In order to constructthe most general possible Lagrangian, one must solve thishuge system!Fortunately, there are some auxiliary conditions that canbe imposed in order to help in solving the Helmholtz con-ditions. First, (H1) is a purely algebraic condition on theunknown matrix g. Auxiliary algebraic conditions on gcan be given as follows: define functions
Ψjki
by
Ψijk =
1
3
(∂Φi
j
∂vk− ∂Φi
k
∂vj
).
The auxiliary condition on g is then
gmiΨmjk+gmkΨ
mij+gmjΨ
mki = 0 for 1 ≤ i, j ≤ n. (A)
In fact, the equations (H2) and (A) are just the first inan infinite hierarchy of similar algebraic conditions. Inthe case of a parallel connection (such as the canonicalconnection on a Lie group), the higher order conditionsare always satisfied, so only (H2) and (A) are of interest.Note that (A) comprises
(n3
)conditions whereas (H1) comprises
(n2
)conditions. Thus, it is possible that (H1) and (A) to-gether imply that the Lagrangian function is singular. Asof 2006, there is no general theorem to circumvent thisdifficulty in arbitrary dimension, although certain specialcases have been resolved.A second avenue of attack is to see whether the system(E) admits a submersion onto a lower-dimensional systemand to try to “lift” a Lagrangian for the lower-dimensionalsystem up to the higher-dimensional one. This is not re-ally an attempt to solve the Helmholtz conditions so muchas it is an attempt to construct a Lagrangian and then showthat its Euler–Lagrange equations are indeed the system(E).
16.3 References• Douglas, Jesse (1941). “Solution of the inverseproblem in the calculus of variations”. Transac-tions of the American Mathematical Society (Trans-actions of the American Mathematical Society, Vol.50, No. 1) 50 (1): 71–128. doi:10.2307/1989912.ISSN 0002-9947. JSTOR 1989912.
• Rawashdeh, M., & Thompson, G. (2006). “The in-verse problem for six-dimensional codimension twonilradical Lie algebras”. Journal of MathematicalPhysics 47 (11): 112901. doi:10.1063/1.2378620.ISSN 0022-2488.
Chapter 17
Jacobi coordinates
Jacobi coordinates for two-body problem; Jacobi coordinates areR = m1
Mx1 +
m2M
x2 and r = x1 −x2 withM = m1 +m2
.[1]
m2
0
m1
m4
m3
x2
x1
r1
x4
x3
r3 R
r2
A possible set of Jacobi coordinates for four-body problem; theJacobi coordinates are r1, r2, r3 and the center of mass R. SeeCornille.[2]
In the theory of many-particle systems, Jacobi coordi-nates often are used to simplify the mathematical for-mulation. These coordinates are particularly common intreating polyatomic molecules and chemical reactions,[3]and in celestial mechanics.[4] An algorithm for generat-ing the Jacobi coordinates for N bodies may be basedupon binary trees.[5] In words, the algorithm is describedas follows:[5]
Letmj andmk be the masses of two bodies
that are replaced by a new body of virtual massM = mj + mk. The position coordinates xj andxk are replaced by their relative position rjk =xj − xk and by the vector to their center of massRjk = (mj qj + mkqk)/(mj + mk). The nodein the binary tree corresponding to the virtualbody has mj as its right child and mk as its leftchild. The order of children indicates the rel-ative coordinate points from xk to xj. Repeatthe above step for N − 1 bodies, that is, the N− 2 original bodies plus the new virtual body.
For the four-body problem the result is:[2]
r1 = x1 − x2 ,
rj =1
m0j
j∑k=1
mkxk − xj+1 ,
with
m0j =
j∑k=1
mk .
The vector R is the center of mass of all the bodies:
R = 1m0
∑Nk=1 mkxk ; m0 =
∑Nk=1 mk .
The result one is left with is thus a system of Ntranslationally-invariant coordinates and a reduced mass,from iteratively treats and reducing two-body systemswithin the many-body system.
17.1 References[1] David Betounes (2001). Differential Equations. Springer.
p. 58; Figure 2.15. ISBN 0-387-95140-7.
[2] Patrick Cornille (2003). “Partition of forces using Ja-cobi coordinates”. Advanced electromagnetism and vac-uum physics. World Scientific. p. 102. ISBN 981-238-367-0.
66
17.1. REFERENCES 67
[3] John Z. H. Zhang (1999). Theory and application ofquantum molecular dynamics. World Scientific. p. 104.ISBN 981-02-3388-4.
[4] For example, see Edward Belbruno (2004). CaptureDynamics and Chaotic Motions in Celestial Mechanics.Princeton University Press. p. 9. ISBN 0-691-09480-2.
[5] Hildeberto Cabral, Florin Diacu (2002). “AppendixA: Canonical transformations to Jacobi coordinates”.Classical and celestial mechanics. Princeton UniversityPress. p. 230. ISBN 0-691-05022-8.
Chapter 18
Joseph-Louis Lagrange
“Lagrange” redirects here. For other uses, see Lagrange(disambiguation).
Joseph-Louis Lagrange (/ləˈɡrɑːndʒ/[1] or/ləˈɡreɪndʒ/;[2] French: [laˈgrɑʒ]), born GiuseppeLodovico Lagrangia[3] or Giuseppe Ludovico Dela Grange Tournier[4] (also reported as GiuseppeLuigi Lagrange[5] or Lagrangia[6]) (25 January 1736– 10 April 1813) was an Italian Enlightenment Eramathematician and astronomer. He made significantcontributions to the fields of analysis, number theory,and both classical and celestial mechanics.In 1766, on the recommendation of Euler andd'Alembert, Lagrange succeeded Euler as the di-rector of mathematics at the Prussian Academy ofSciences in Berlin, Prussia, where he stayed for overtwenty years, producing volumes of work and winningseveral prizes of the French Academy of Sciences.Lagrange’s treatise on analytical mechanics (MécaniqueAnalytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils,1888–89), written in Berlin and first published in 1788,offered the most comprehensive treatment of classicalmechanics since Newton and formed a basis for thedevelopment of mathematical physics in the nineteenthcentury.In 1787, at age 51, he moved from Berlin to Paris andbecame a member of the French Academy. He remainedin France until the end of his life. He was significantlyinvolved in the decimalisation in Revolutionary France,became the first professor of analysis at the École Poly-technique upon its opening in 1794, founding member ofthe Bureau des Longitudes and Senator in 1799.
18.1 Scientific contribution
Lagrange was one of the creators of the calculus of varia-tions, deriving the Euler–Lagrange equations for extremaof functionals. He also extended the method to take intoaccount possible constraints, arriving at the method ofLagrange multipliers. Lagrange invented the method ofsolving differential equations known as variation of pa-rameters, applied differential calculus to the theory of
probabilities and attained notable work on the solution ofequations. He proved that every natural number is a sumof four squares. His treatise Theorie des fonctions analy-tiques laid some of the foundations of group theory, antic-ipating Galois. In calculus, Lagrange developed a novelapproach to interpolation and Taylor series. He studiedthe three-body problem for the Earth, Sun and Moon(1764) and the movement of Jupiter’s satellites (1766),and in 1772 found the special-case solutions to this prob-lem that yield what are now known as Lagrangian points.But above all, he is best known for his work on mechan-ics, where he has transformed Newtonian mechanics intoa branch of analysis, Lagrangian mechanics as it is nowcalled, and presented the so-called mechanical “princi-ples” as simple results of the variational calculus.
18.2 Biography
In appearance he was of medium height, and slightlyformed, with pale blue eyes and a colourless complexion.In character he was nervous and timid, he detested con-troversy, and to avoid it willingly allowed others to takethe credit for what he had himself done.He always thought out the subject of his papers before hebegan to compose them, and usually wrote them straightoff without a single erasure or correction.W.W. Rouse Ball[7]
18.2.1 Early years
Born as Giuseppe Lodovico Lagrangia, Lagrange wasof Italian and French descent. His paternal great-grandfather was a French army officer who had moved toTurin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange’s time, and married an Italian; sodid his grandfather and his father. His mother was fromthe countryside of Turin.[8] He was raised as a RomanCatholic (but later on became an agnostic).[9]
His father, who had charge of the king’s military chest andwas Treasurer of the Office of Public Works and Forti-fications in Turin, should have maintained a good social
68
18.2. BIOGRAPHY 69
position and wealth, but before his son grew up he had lostmost of his property in speculations. A career as a lawyerwas planned out for Lagrange by his father, and certainlyLagrange seems to have accepted this willingly. He stud-ied at the College of Turin and his favourite subject wasclassical Latin. At first he had no great enthusiasm formathematics, finding Greek geometry rather dull.It was not until he was seventeen that he showed any tastefor mathematics – his interest in the subject being first ex-cited by a paper by Edmond Halley which he came acrossby accident. Alone and unaided he threw himself intomathematical studies; at the end of a year’s incessant toilhe was already an accomplished mathematician. CharlesEmmanuel III appointed Lagrange to serve as the “Sosti-tuto del Maestro di Matematica” (mathematics assistantprofessor) at the Royal Military Academy of the The-ory and Practice of Artillery in 1755, where he taughtcourses in calculus and mechanics to support the Pied-montese army’s early adoption of the ballistics theories ofBenjamin Robins and Leonhard Euler. In that capacity,Lagrange was the first to teach calculus in an engineer-ing school. According to Alessandro Papacino D'Antoni,the academy’s military commander and famous artillerytheorist, Lagrange unfortunately proved to be a problem-atic professor with his oblivious teaching style, abstractreasoning, and impatience with artillery and fortification-engineering applications.[10] In this Academy one of hisstudents was François Daviet de Foncenex.[11]
Variational calculus
Lagrange is one of the founders of the calculus of vari-ations. Starting in 1754, he worked on the problemof tautochrone, discovering a method of maximizingand minimizing functionals in a way similar to findingextrema of functions. Lagrange wrote several lettersto Leonhard Euler between 1754 and 1756 describinghis results. He outlined his "δ-algorithm”, leading tothe Euler–Lagrange equations of variational calculus andconsiderably simplifying Euler’s earlier analysis.[12] La-grange also applied his ideas to problems of classical me-chanics, generalizing the results of Euler and Maupertuis.Euler was very impressed with Lagrange’s results. It hasbeen stated that “with characteristic courtesy he with-held a paper he had previously written, which coveredsome of the same ground, in order that the young Ital-ian might have time to complete his work, and claim theundisputed invention of the new calculus"; however, thischivalric view has been disputed.[13] Lagrange publishedhis method in two memoirs of the Turin Society in 1762and 1773.
Miscellanea Taurinensia
In 1758, with the aid of his pupils (mainly Daviet deFoncenex), Lagrange established a society, which was
subsequently incorporated as the Turin Academy of Sci-ences, and most of his early writings are to be found inthe five volumes of its transactions, usually known as theMiscellanea Taurinensia. Many of these are elaborate pa-pers. The first volume contains a paper on the theory ofthe propagation of sound; in this he indicates a mistakemade byNewton, obtains the general differential equationfor the motion, and integrates it for motion in a straightline. This volume also contains the complete solution ofthe problem of a string vibrating transversely; in this pa-per he points out a lack of generality in the solutions pre-viously given by Brook Taylor, D'Alembert, and Euler,and arrives at the conclusion that the form of the curve atany time t is given by the equation y = a sin(mx) sin(nt). The article concludes with a masterly discussion ofechoes, beats, and compound sounds. Other articles inthis volume are on recurring series, probabilities, and thecalculus of variations.The second volume contains a long paper embodying theresults of several papers in the first volume on the theoryand notation of the calculus of variations; and he illus-trates its use by deducing the principle of least action,and by solutions of various problems in dynamics.The third volume includes the solution of several dynami-cal problems by means of the calculus of variations; somepapers on the integral calculus; a solution of Fermat'sproblem mentioned above: given an integer n which isnot a perfect square, to find a number x such that x2n + 1is a perfect square; and the general differential equationsof motion for three bodies moving under their mutual at-tractions.The next work he produced was in 1764 on the librationof the Moon, and an explanation as to why the sameface was always turned to the earth, a problem which hetreated by the aid of virtual work. His solution is espe-cially interesting as containing the germ of the idea ofgeneralized equations of motion, equations which he firstformally proved in 1780.
18.2.2 Berlin
Already in 1756, Euler and Maupertuis, seeing his math-ematical talent, tried to persuade him to come to Berlin,but Lagrange had no such intention and shyly refused theoffer. In 1765, d'Alembert interceded on Lagrange’s be-half with Frederick of Prussia and by letter, asked himto leave Turin for a considerably more prestigious posi-tion in Berlin. Lagrange again turned down the offer, re-sponding that[14]:361
It seems to me that Berlin would not be at allsuitable for me while M.Euler is there.
In 1766, Euler left Berlin for Saint Petersburg, and Fred-erick himself wrote to Lagrange expressing the wish of“the greatest king in Europe” to have “the greatest math-
70 CHAPTER 18. JOSEPH-LOUIS LAGRANGE
ematician in Europe” resident at his court. Lagrange wasfinally persuaded and he spent the next twenty years inPrussia, where he produced not only the long series ofpapers published in the Berlin and Turin transactions, butalso his monumental work, the Mécanique analytique. In1767, he married his cousin Vittoria Conti.Lagrange was a favourite of the king, who used frequentlyto discourse to him on the advantages of perfect regularityof life. The lesson went home, and thenceforth Lagrangestudied his mind and body as though they were machines,and found by experiment the exact amount of work whichhe was able to do without breaking down. Every night heset himself a definite task for the next day, and on com-pleting any branch of a subject he wrote a short analysisto see what points in the demonstrations or in the subject-matter were capable of improvement. He always thoughtout the subject of his papers before he began to composethem, and usually wrote them straight off without a singleerasure or correction.Nonetheless, during his years in Berlin, Lagrange’s healthwas rather poor on many occasions, and that of his wifeVittoria was even worse. She died in 1783 after years ofillness and Lagrange was very depressed. In 1786, Fred-erick II died, and the climate of Berlin became rather try-ing for Lagrange.[8]
18.2.3 Paris
In 1786, following Frederick’s death, Lagrange re-ceived similar invitations from states including Spain andNaples, and he accepted the offer of Louis XVI to moveto Paris. In France he was received with every mark ofdistinction and special apartments in the Louvre were pre-pared for his reception, and he became a member of theFrench Academy of Sciences, which became part of theInstitut de France (1795). At the beginning of his resi-dence in Paris he was seized with an attack ofmelancholy,and even the printed copy of his Mécanique on which hehad worked for a quarter of a century lay for more thantwo years unopened on his desk. Curiosity as to the re-sults of the French revolution first stirred him out of hislethargy, a curiosity which soon turned to alarm as therevolution developed.It was about the same time, 1792, that the unaccountablesadness of his life and his timidity moved the compas-sion of 24-year-old Renée-Françoise-Adélaïde Le Mon-nier, daughter of his friend, the astronomer Pierre CharlesLe Monnier. She insisted on marrying him, and proved adevoted wife to whom he became warmly attached.In September of 1793, the Reign of Terror began. Un-der intervention of Antoine Lavoisier, who himself wasby then already thrown out of the Academy along withmany other scholars, Lagrange was specifically exemptedby name in the decree of October 1793 that ordered allforeigners to leave France. On May 4, 1794, Lavoisierand 27 other tax farmers were arrested and sentenced to
death and guillotined on the afternoon after the trial. La-grange said on the death of Lavoisier:
It took only a moment to cause this head to falland a hundred years will not suffice to produceits like.[8]
Though Lagrange had been preparing to escape fromFrance while there was yet time, he was never in any dan-ger; different revolutionary governments (and at a latertime, Napoleon) loaded him with honors and distinctions.This luckiness or safety may to some extent be due to hislife attitude he expressed many years before: "I believethat, in general, one of the first principles of every wiseman is to conform strictly to the laws of the country inwhich he is living, even when they are unreasonable".[8] Astriking testimony to the respect in which he was held wasshown in 1796 when the French commissary in Italy wasordered to attend in full state on Lagrange’s father, andtender the congratulations of the republic on the achieve-ments of his son, who “had done honor to all mankind byhis genius, and whom it was the special glory of Piedmontto have produced.” It may be added that Napoleon, whenhe attained power, warmly encouraged scientific studiesin France, and was a liberal benefactor of them. Ap-pointed senator in 1799, he was the first signer of theSénatus-consulte which in 1802 annexed his fatherlandPiedmont to France.[5] He acquired French citizenship inconsequence.[5]
Units of measurement
Lagrange was considerably involved in the process ofmaking new standard units of measurement in the 1790s.He was offered the presidency of the Commission for thereform of weights and measures (la Commission des Poidset Mesures) when he was preparing to escape. And af-ter Lavoisier’s death in 1794, it was largely owing to La-grange’s influence that the final choice of the unit systemof metre and kilogram was settled and the decimal sub-division was finally accepted by the commission of 1799.Lagrange was also one of the founding members of theBureau des Longitudes in 1795.
École normale
In 1795, Lagrange was appointed to a mathematical chairat the newly established École normale, which enjoyedonly a brief existence of four months. His lectures therewere quite elementary, and contain nothing of any specialimportance, but they were published because the profes-sors had to “pledge themselves to the representatives ofthe people and to each other neither to read nor to re-peat from memory,” and the discourses were ordered tobe taken down in shorthand in order to enable the deputiesto see how the professors acquitted themselves.
18.3. WORK IN BERLIN 71
École Polytechnique
In 1794, Lagrange was appointed professor of the ÉcolePolytechnique; and his lectures there, described by math-ematicians who had the good fortune to be able to attendthem, were almost perfect both in form and matter. Be-ginning with the merest elements, he led his hearers onuntil, almost unknown to themselves, they were them-selves extending the bounds of the subject: above all heimpressed on his pupils the advantage of always usinggeneral methods expressed in a symmetrical notation.But Lagrange does not seem to have been a success-ful teacher. Fourier, who attended his lectures in 1795,wrote:
his voice is very feeble, at least in that he doesnot become heated; he has a very marked Ital-ian accent and pronounces the s like z [...] Thestudents, of whom the majority are incapableof appreciating him, give him little welcome,but the professeurs make amends for it.[15]
Late years
Lagrange’s tomb in the crypt of the Panthéon
In 1810, Lagrange commenced a thorough revision of theMécanique analytique, but he was able to complete onlyabout two-thirds of it before his death at Paris in 1813,in 128 Rue du Faubourg Saint Honoré. Napoleon hon-oured him with the Grand Croix of the Ordre Impérial dela Réunion just two days before he died. He was buriedthat same year in the Panthéon in Paris. The French in-scription on his tomb there reads:
JOSEPH LOUIS LAGRANGE. Senator.Count of the Empire. Grand Officer of the Le-gion of Honour. Grand Cross of the ImperialOrder of the Reunion. Member of the Instituteand the Bureau of Longitude. Born in Turinon 25 January 1736. Died in Paris on 10 April1813.
18.3 Work in Berlin
Lagrange was extremely active scientifically duringtwenty years he spent in Berlin. Not only did he pro-duce his Mécanique analytique, but he contributed be-tween one and two hundred papers to the Academy ofTurin, the Berlin Academy, and the French Academy.Some of these are really treatises, and all without excep-tion are of a high order of excellence. Except for a shorttime when he was ill he produced on average about onepaper a month. Of these, note the following as amongstthe most important.First, his contributions to the fourth and fifth volumes,1766–1773, of theMiscellanea Taurinensia; of which themost important was the one in 1771, in which he dis-cussed how numerous astronomical observations shouldbe combined so as to give the most probable result. Andlater, his contributions to the first two volumes, 1784–1785, of the transactions of the Turin Academy; to thefirst of which he contributed a paper on the pressure ex-erted by fluids in motion, and to the second an article onintegration by infinite series, and the kind of problems forwhich it is suitable.Most of the papers sent to Paris were on astronomi-cal questions, and among these one ought to particularlymention his paper on the Jovian system in 1766, his essayon the problem of three bodies in 1772, his work on thesecular equation of the Moon in 1773, and his treatise oncometary perturbations in 1778. These were all writtenon subjects proposed by the Académie française, and ineach case the prize was awarded to him.
18.3.1 Lagrangian mechanics
Between 1772 and 1788, Lagrange re-formulated Classi-cal/Newtonian mechanics to simplify formulas and easecalculations. These mechanics are called Lagrangian me-chanics.
18.3.2 Algebra
The greater number of his papers during this time were,however, contributed to the Prussian Academy of Sci-ences. Several of them deal with questions in algebra.
• His discussion of representations of integers byquadratic forms (1769) and by more general alge-braic forms (1770).
• His tract on the Theory of Elimination, 1770.
• Lagrange’s theorem that the order of a subgroup Hof a group G must divide the order of G.
• His papers of 1770 and 1771 on the general processfor solving an algebraic equation of any degree via
72 CHAPTER 18. JOSEPH-LOUIS LAGRANGE
the Lagrange resolvents. This method fails to givea general formula for solutions of an equation ofdegree five and higher, because the auxiliary equa-tion involved has higher degree than the original one.The significance of this method is that it exhibitsthe already known formulas for solving equations ofsecond, third, and fourth degrees as manifestationsof a single principle, and was foundational in Galoistheory. The complete solution of a binomial equa-tion of any degree is also treated in these papers.
• In 1773, Lagrange considered a functional deter-minant of order 3, a special case of a Jacobian.He also proved the expression for the volume of atetrahedron with one of the vertices at the origin asthe one sixth of the absolute value of the determinantformed by the coordinates of the other three ver-tices.
18.3.3 Number theory
Several of his early papers also deal with questions ofnumber theory.
• Lagrange (1766–1769) was the first to prove thatPell’s equation x2 − ny2 = 1 has a nontrivial solu-tion in the integers for any non-square natural num-ber n.[16]
• He proved the theorem, stated by Bachet withoutjustification, that every positive integer is the sumof four squares, 1770.
• He proved Wilson’s theorem that (for any integern>1) n is a prime if and only if (n − 1)! + 1 is amultiple of n, 1771.
• His papers of 1773, 1775, and 1777 gave demon-strations of several results enunciated by Fermat,and not previously proved.
• His Recherches d'Arithmétique of 1775 developeda general theory of binary quadratic forms to han-dle the general problem of when an integer is repre-sentable by the form ax2 + by2 + cxy .
• He made contributions to the theory of continuedfractions.
18.3.4 Other mathematical work
There are also numerous articles on various points ofanalytical geometry. In two of them, written ratherlater, in 1792 and 1793, he reduced the equations of thequadrics (or conicoids) to their canonical forms.During the years from 1772 to 1785, he contributed along series of papers which created the science of partialdifferential equations. A large part of these results were
collected in the second edition of Euler’s integral calculuswhich was published in 1794.
18.3.5 Astronomy
Lastly, there are numerous papers on problems inastronomy. Of these the most important are the follow-ing:
• Attempting to solve the general three-body problem,with the consequent discovery of the two constant-pattern solutions, collinear and equilateral, 1772.Those solutions were later seen to explain what arenow known as the Lagrangian points.
• On the attraction of ellipsoids, 1773: this is foundedon Maclaurin's work.
• On the secular equation of the Moon, 1773; also no-ticeable for the earliest introduction of the idea ofthe potential. The potential of a body at any pointis the sum of the mass of every element of the bodywhen divided by its distance from the point. La-grange showed that if the potential of a body at anexternal point were known, the attraction in any di-rection could be at once found. The theory of thepotential was elaborated in a paper sent to Berlin in1777.
• On the motion of the nodes of a planet’s orbit, 1774.
• On the stability of the planetary orbits, 1776.
• Two papers in which the method of determining theorbit of a comet from three observations is com-pletely worked out, 1778 and 1783: this has not in-deed proved practically available, but his system ofcalculating the perturbations by means of mechani-cal quadratures has formed the basis of most subse-quent researches on the subject.
• His determination of the secular and periodic vari-ations of the elements of the planets, 1781–1784:the upper limits assigned for these agree closely withthose obtained later by Le Verrier, and Lagrangeproceeded as far as the knowledge then possessedof the masses of the planets permitted.
• Three papers on the method of interpolation, 1783,1792 and 1793: the part of finite differences dealingtherewith is now in the same stage as that in whichLagrange left it.
18.3.6 Mécanique analytique
Over and above these various papers he composed hisgreat treatise, the Mécanique analytique. In this he laysdown the law of virtual work, and from that one funda-mental principle, by the aid of the calculus of variations,deduces the whole of mechanics, both of solids and fluids.
18.4. WORK IN FRANCE 73
The object of the book is to show that the subject is im-plicitly included in a single principle, and to give gen-eral formulae from which any particular result can be ob-tained. The method of generalized co-ordinates by whichhe obtained this result is perhaps the most brilliant re-sult of his analysis. Instead of following the motion ofeach individual part of a material system, as D'Alembertand Euler had done, he showed that, if we determine itsconfiguration by a sufficient number of variables whosenumber is the same as that of the degrees of freedompossessed by the system, then the kinetic and potentialenergies of the system can be expressed in terms of thosevariables, and the differential equations of motion thencededuced by simple differentiation. For example, in dy-namics of a rigid system he replaces the consideration ofthe particular problem by the general equation, which isnow usually written in the form
d
dt
∂T
∂θ− ∂T
∂θ+
∂V
∂θ= 0,
where T represents the kinetic energy and V representsthe potential energy of the system. He then presentedwhat we now know as the method of Lagrange mul-tipliers—though this is not the first time that methodwas published—as a means to solve this equation.[17]Amongst other minor theorems here given it may men-tion the proposition that the kinetic energy imparted bythe given impulses to a material system under given con-straints is a maximum, and the principle of least action.All the analysis is so elegant that Sir William RowanHamilton said the work could be described only as a sci-entific poem. Lagrange remarked that mechanics was re-ally a branch of pure mathematics analogous to a geom-etry of four dimensions, namely, the time and the threecoordinates of the point in space; and it is said that heprided himself that from the beginning to the end of thework there was not a single diagram. At first no printercould be foundwhowould publish the book; but Legendreat last persuaded a Paris firm to undertake it, and it was is-sued under the supervision of Laplace, Cousin, Legendre(editor) and Condorcet in 1788.[8]
18.4 Work in France
18.4.1 Differential calculus and calculus ofvariations
Lagrange’s lectures on the differential calculus at ÉcolePolytechnique form the basis of his treatise Théorie desfonctions analytiques, which was published in 1797. Thiswork is the extension of an idea contained in a paper hehad sent to the Berlin papers in 1772, and its object is tosubstitute for the differential calculus a group of theoremsbased on the development of algebraic functions in series,relying in particular on the principle of the generality of
algebra.A somewhat similar method had been previously usedby John Landen in the Residual Analysis, published inLondon in 1758. Lagrange believed that he could thusget rid of those difficulties, connected with the use ofinfinitely large and infinitely small quantities, to whichphilosophers objected in the usual treatment of the dif-ferential calculus. The book is divided into three parts:of these, the first treats of the general theory of functions,and gives an algebraic proof of Taylor’s theorem, the va-lidity of which is, however, open to question; the seconddeals with applications to geometry; and the third withapplications to mechanics.Another treatise on the same lines was his Leçons surle calcul des fonctions, issued in 1804, with the secondedition in 1806. It is in this book that Lagrange formu-lated his celebrated method of Lagrange multipliers, inthe context of problems of variational calculus with in-tegral constraints. These works devoted to differentialcalculus and calculus of variations may be considered asthe starting point for the researches of Cauchy, Jacobi,and Weierstrass.
18.4.2 Infinitesimals
At a later period Lagrange fully embraced the use ofinfinitesimals in preference to founding the differentialcalculus on the study of algebraic forms; and in the pref-ace to the second edition of the Mécanique Analytique,which was issued in 1811, he justifies the employment ofinfinitesimals, and concludes by saying that:
When we have grasped the spirit of the infinites-imal method, and have verified the exactness ofits results either by the geometrical method ofprime and ultimate ratios, or by the analyticalmethod of derived functions, we may employ in-finitely small quantities as a sure and valuablemeans of shortening and simplifying our proofs.
18.4.3 Number theory
His Résolution des équations numériques, published in1798, was also the fruit of his lectures at École Polytech-nique. There he gives the method of approximating tothe real roots of an equation by means of continued frac-tions, and enunciates several other theorems. In a note atthe end he shows how Fermat’s little theorem, that is
ap−1 − 1 ≡ 0 (mod p)
where p is a prime and a is prime to p, may be appliedto give the complete algebraic solution of any binomialequation. He also here explains how the equation whoseroots are the squares of the differences of the roots of the
74 CHAPTER 18. JOSEPH-LOUIS LAGRANGE
original equation may be used so as to give considerableinformation as to the position and nature of those roots.
18.4.4 Celestial mechanics
The theory of the planetary motions had formed the sub-ject of some of the most remarkable of Lagrange’s Berlinpapers. In 1806 the subject was reopened by Poisson,who, in a paper read before the French Academy, showedthat Lagrange’s formulae led to certain limits for the sta-bility of the orbits. Lagrange, who was present, now dis-cussed the whole subject afresh, and in a letter commu-nicated to the Academy in 1808 explained how, by thevariation of arbitrary constants, the periodical and secularinequalities of any system of mutually interacting bodiescould be determined.
18.5 Prizes and distinctions
Euler proposed Lagrange for election to the BerlinAcademy and he was elected on 2 September 1756. Hewas elected a Fellow of the Royal Society of Edinburgh in1790, a Fellow of the Royal Society and a foreign mem-ber of the Royal Swedish Academy of Sciences in 1806.In 1808, Napoleon made Lagrange a Grand Officer ofthe Legion of Honour and a Count of the Empire. Hewas awarded the Grand Croix of the Ordre Impérial de laRéunion in 1813, a week before his death in Paris.Lagrange was awarded the 1764 prize of the FrenchAcademy of Sciences for his memoir on the libration ofthe Moon. In 1766 the Academy proposed a problemof the motion of the satellites of Jupiter, and the prizeagain was awarded to Lagrange. He also shared or wonthe prizes of 1772, 1774, and 1778.Lagrange is one of the 72 prominent French scientistswho were commemorated on plaques at the first stage ofthe Eiffel Tower when it first opened. Rue Lagrange inthe 5th Arrondissement in Paris is named after him. InTurin, the street where the house of his birth still standsis named via Lagrange. The lunar crater Lagrange alsobears his name.
18.6 See also
• List of things named after Joseph-Louis Lagrange
18.7 Notes
The initial version of this article was taken from the publicdomain resource A Short Account of the History of Math-ematics (4th edition, 1908) by W. W. Rouse Ball.
[1] “Lagrange”. Merriam-Webster Dictionary.
[2] “Lagrange”. Random House Webster’s Unabridged Dictio-nary.
[3] Briano, Giorgio (1861),Giuseppe Luigi Lagrangia (in Ital-ian), Torino: Unione Tipografica Editrice
[4] Angelo Genocchi. “Luigi Lagrange” (PDF). Il primo sec-olo della R. Accademia delle Scienze di Torino (in Italian).Accademia delle Scienze di Torino. pp. 86–95. Retrieved2 January 2014.
[5] Luigi Pepe. “Giuseppe Luigi Lagrange”. Dizionario Bi-ografico degli Italiani (in Italian). Enciclopedia Italiana.Retrieved 8 July 2012.
[6] Encyclopedia of Space and Astronomy.
[7] W. W. Rouse Ball, 1908, Joseph Louis Lagrange (1736–1813)," A Short Account of the History of Mathematics,4th ed. pp. 401–412. Complete article online, p.338 and333:
[8] Lagrange St. Andrew University
[9] Morris Kline (1986). Mathematics and the Search forKnowledge. Oxford University Press. p. 214. ISBN978-0-19-504230-6. Lagrange and Laplace, though ofCatholic parentage, were agnostics.
[10] Steele, Brett (2005). “13”. In Brett Steele and TameraDorland. The Heirs of Archimedes: Science and the Art ofWar through the Age of Enlightenment. Cambridge: MITPress. pp. 368, 375. ISBN 0-262-19516-X.
[11] de Andrade Martins, Roberto (2008). “A busca da Ciên-cia a priori no final do Seculo XVIII e a origem da Análisedimensional”. In Roberto de Andrade Martins, Lilian Al-Chueyr Pereira Martins, Cibelle Celestino Silva, JulianaMesquita Hidalgo Ferreira (eds.). Filosofia E Historia DaCiência No Cone Sul. 3 Encontro (in Portuguese). AFHIC.p. 406. ISBN 978-1-4357-1633-9.
[12] Although some authors speak of general method ofsolving "isoperimetric problems”, the eighteenth centurymeaning of this expression amounts to “problems in vari-ational calculus”, reserving the adjective “relative” forproblems with isoperimetric-type constraints. The cele-brated method of Lagrange multipliers, which applies tooptimization of functions of several variables subject toconstraints, did not appear until much later. See Fraser,Craig (1992). “Isoperimetric Problems in the VariationalCalculus of Euler and Lagrange”. Historia Mathematica19: 4–23. doi:10.1016/0315-0860(92)90052-D.
[13] Galletto, D., The genesis of Mécanique analytique, LaMé-canique analytique de Lagrange et son héritage, II (Turin,1989). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.126 (1992), suppl. 2, 277–370, MR 1264671.
[14] Richard B. Vinter (2000). Optimal Control. Springer.ISBN 978-0-8176-4075-0.
[15] Ivor Grattan-Guiness. Convolutions in French Mathemat-ics, 1800-1840. Birkhäuser 1990. Vol. I, p.108.
[16] Oeveres, t.1, 671–732
18.9. EXTERNAL LINKS 75
[17] Marco Panza, “The Origins of Analytic Mechanics in the18th Century”, in Hans Niels Jahnke (editor), A Historyof Analysis, 2003, p. 149
18.8 References• Maria Teresa Borgato, Luigi Pepe (1990), La-grange, appunti per una biografia scientifica (in Ital-ian), Torino: La Rosa
• Columbia Encyclopedia, 6th ed., 2005, "Lagrange,Joseph Louis."
• W. W. Rouse Ball, 1908, "Joseph Louis Lagrange(1736–1813)" A Short Account of the History ofMathematics, 4th ed. also on Gutenberg
• Chanson, Hubert, 2007, "Velocity Potential in RealFluid Flows: Joseph-Louis Lagrange’s Contribu-tion," La Houille Blanche 5: 127–31.
• Fraser, Craig G., 2005, “Théorie des fonctions an-alytiques” in Grattan-Guinness, I., ed., LandmarkWritings in Western Mathematics. Elsevier: 258–76.
• Lagrange, Joseph-Louis. (1811). Mécanique Ana-lytique. Courcier (reissued by Cambridge UniversityPress, 2009; ISBN 978-1-108-00174-8)
• Lagrange, J.L. (1781) “Mémoire sur la Théorie duMouvement des Fluides"(Memoir on the Theory ofFluid Motion) in Serret, J.A., ed., 1867. Oeuvresde Lagrange, Vol. 4. Paris” Gauthier-Villars: 695–748.
• Pulte, Helmut, 2005, “Méchanique Analytique” inGrattan-Guinness, I., ed., Landmark Writings inWestern Mathematics. Elsevier: 208–24.
• A. Conte, C. Mancinelli, E. Borgi, L. Pepe (editors)(2013), Lagrange. Un europeo a Torino (in Italian),Torino: Hapax Editore, ISBN 978-88-88000-57-2
18.9 External links• O'Connor, John J.; Robertson, Edmund F., “Joseph-Louis Lagrange”,MacTutor History of Mathematicsarchive, University of St Andrews.
• Weisstein, Eric W., Lagrange, Joseph (1736-1813)from ScienceWorld.
• Lagrange, Joseph Louis de: The Encyclopedia ofAstrobiology, Astronomy and Space Flight
• Joseph-Louis Lagrange at the Mathematics Geneal-ogy Project
• The Founders of Classical Mechanics: Joseph LouisLagrange
• The Lagrange Points
• Derivation of Lagrange’s result (not Lagrange’smethod)
• Lagrange’s works (in French) Oeuvres de Lagrange,edited by Joseph Alfred Serret, Paris 1867, digitizedby Göttinger Digitalisierungszentrum (Mécaniqueanalytique is in volumes 11 and 12.)
• Joseph Louis de Lagrange – Œuvres complètesGallica-Math
• Inventaire chronologique de l'œuvre de LagrangePersee
• Works by Joseph-Louis Lagrange at Project Guten-berg
• Works by or about Joseph-Louis Lagrange atInternet Archive
Chapter 19
Lagrangian point
“Lagrange Point” redirects here. For the video game, seeLagrange Point (video game).This article is about three-body libration points. Fortwo-body libration points, see Geostationary orbit#Earthorbital libration points.
In celestial mechanics, the Lagrangian points(/ləˈɡrɑːndʒiən/; also Lagrange points, L-points,or libration points) are positions in an orbital configu-ration of two large bodies where a small object affectedonly by gravity can maintain a stable position relativeto the two large bodies. The Lagrange points markpositions where the combined gravitational pull of thetwo large masses provides precisely the centripetal forcerequired to orbit with them. There are five such points,labeled L1 to L5, all in the orbital plane of the two largebodies. The first three are on the line connecting the twolarge bodies and the last two, L4 and L5, each form anequilateral triangle with the two large bodies. The twolatter points are stable, which implies that objects canorbit around them in a rotating coordinate system tied tothe two large bodies.Several planets have minor planets near their L4 and L5points (trojans) with respect to the Sun, with Jupiter inparticular having more than a million of these. Artificialsatellites have been placed at L1 and L2 with respect tothe Sun and Earth, and Earth and the Moon for variouspurposes, and the Lagrangian points have been proposedfor a variety of future uses in space exploration.
19.1 History
The three collinear Lagrange points (L1, L2, L3) werediscovered by Leonhard Euler a few years before La-grange discovered the remaining two.[1][2]
In 1772, Joseph-Louis Lagrange published an “Essay onthe three-body problem". In the first chapter he consid-ered the general three-body problem. From that, in thesecond chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for anythree masses, with circular orbits.[3]
L1 L2L3
L4
L5
Lagrange points in the Sun–Earth system (not to scale)
19.2 Lagrange points
The five Lagrangian points are labeled and defined as fol-lows:The L1 point lies on the line defined by the two largemasses M1 and M2, and between them. It is the mostintuitively understood of the Lagrangian points: the onewhere the gravitational attraction of M2 partially cancelsM1's gravitational attraction.
Explanation: An object that orbits the Sunmore closely than Earth would normally havea shorter orbital period than Earth, but that ig-nores the effect of Earth’s own gravitationalpull. If the object is directly between Earth andthe Sun, then Earth’s gravity counteracts someof the Sun’s pull on the object, and thereforeincreases the orbital period of the object. Thecloser to Earth the object is, the greater thiseffect is. At the L1 point, the orbital period ofthe object becomes exactly equal to Earth’s or-bital period. L1 is about 1.5 million kilometersfrom Earth.[4]
TheL2 point lies on the line through the two largemasses,beyond the smaller of the two. Here, the gravitational
76
19.3. NATURAL OBJECTS AT LAGRANGIAN POINTS 77
forces of the two large masses balance the centrifugal ef-fect on a body at L2.
Explanation: On the opposite side of Earthfrom the Sun, the orbital period of an objectwould normally be greater than that of Earth.The extra pull of Earth’s gravity decreases theorbital period of the object, and at the L2 pointthat orbital period becomes equal to Earth’s.Like L1, L2 is about 1.5 million kilometersfrom Earth.
The L3 point lies on the line defined by the two largemasses, beyond the larger of the two.
Explanation: L3 in the Sun–Earth system ex-ists on the opposite side of the Sun, a littleoutside Earth’s orbit but slightly closer to theSun than Earth is. (This apparent contradictionis because the Sun is also affected by Earth’sgravity, and so orbits around the two bodies’barycenter, which is, however, well inside thebody of the Sun.) At the L3 point, the com-bined pull of Earth and Sun again causes theobject to orbit with the same period as Earth.
L
b
E
M
4
Gravitational accelerations at L4
The L4 and L5 points lie at the third corners of the twoequilateral triangles in the plane of orbit whose commonbase is the line between the centers of the two masses,such that the point lies behind (L5) or ahead (L4) of thesmaller mass with regard to its orbit around the largermass.The triangular points (L4 and L5) are stable equilib-ria, provided that the ratio of M1/M2 is greater than24.96.[note 1][5] This is the case for the Sun–Earth sys-tem, the Sun–Jupiter system, and, by a smaller margin,the Earth–Moon system. When a body at these pointsis perturbed, it moves away from the point, but the fac-tor opposite of that which is increased or decreased by
the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bendingthe object’s path into a stable, kidney-bean-shaped orbitaround the point (as seen in the corotating frame of ref-erence).In contrast to L4 and L5, where stable equilibrium exists,the points L1, L2, and L3 are positions of unstable equi-librium. Any object orbiting at one of L1–L3 will tend tofall out of orbit; it is therefore rare to find natural objectsthere, and spacecraft inhabiting these areas must employstation keeping in order to maintain their position.
19.3 Natural objects at Lagrangianpoints
It is common to find objects at or orbiting the L4 and L5
points of natural orbital systems. These are commonlycalled “trojans"; in the 20th century, asteroids discoveredorbiting at the Sun–Jupiter L4 and L5 points were namedafter characters from Homer's Iliad. Asteroids at the L4
point, which leads Jupiter, are referred to as the "Greekcamp", whereas those at the L5 point are referred to asthe "Trojan camp".Other examples of natural objects orbiting at Lagrangepoints:
• The Sun–Earth L4 and L5 points contain interplan-etary dust and at least one asteroid, 2010 TK7, de-tected in October 2010 by Wide-field Infrared Sur-vey Explorer (WISE) and announced during July2011.[6][7]
• The Earth–Moon L4 and L5 points may containinterplanetary dust in what are called Kordylewskiclouds; however, the Hiten spacecraft’s MunichDust Counter (MDC) detected no increase in dustduring its passes through these points. Stability atthese specific points is greatly complicated by solargravitational influence.[8]
• Recent observations suggest that the Sun–NeptuneL4 and L5 points, known as the Neptune trojans,may be very thickly populated,[9] containing largebodies an order of magnitude more numerous thanthe Jupiter trojans.
• Several asteroids also orbit near the Sun-Jupiter L3
point, called the Hilda family.
• The Saturnian moon Tethys has two smaller moonsin its L4 and L5 points, Telesto and Calypso. TheSaturnian moon Dione also has two Lagrangianco-orbitals, Helene at its L4 point and Polydeucesat L5. The moons wander azimuthally about theLagrangian points, with Polydeuces describing thelargest deviations, moving up to 32 degrees awayfrom the Saturn–Dione L5 point. Tethys and Dione
78 CHAPTER 19. LAGRANGIAN POINT
are hundreds of times more massive than their “es-corts” (see the moons’ articles for exact diameter fig-ures; masses are not known in several cases), andSaturn is far more massive still, which makes theoverall system stable.
• One version of the giant impact hypothesis suggeststhat an object named Theia formed at the Sun–EarthL4 or L5 points and crashed into Earth after its orbitdestabilized, forming the Moon.
• Mars has four known co-orbital asteroids (5261 Eu-reka, 1999 UJ7, 1998 VF31 and 2007 NS2), all atits Lagrangian points.
• Earth’s companion object 3753 Cruithne is in a re-lationship with Earth that is somewhat trojan-like,but that is different from a true trojan. Cruithne oc-cupies one of two regular solar orbits, one of themslightly smaller and faster than Earth’s, and the otherslightly larger and slower. It periodically alternatesbetween these two orbits due to close encounterswith Earth. When it is in the smaller, faster orbitand approaches Earth, it gains orbital energy fromEarth and moves up into the larger, slower orbit. Itthen falls farther and farther behind Earth, and even-tually Earth approaches it from the other direction.Then Cruithne gives up orbital energy to Earth, anddrops back into the smaller orbit, thus beginning thecycle anew. The cycle has no noticeable impact onthe length of the year, because Earth’s mass is over20 billion (2×1010) times more than that of 3753Cruithne.
• Epimetheus and Janus, satellites of Saturn, havea similar relationship, though they are of similarmasses and so actually exchange orbits with eachother periodically. (Janus is roughly 4 times moremassive but still light enough for its orbit to be al-tered.) Another similar configuration is known asorbital resonance, in which orbiting bodies tend tohave periods of a simple integer ratio, due to theirinteraction.
• In a binary star system, the Roche lobe has its apexlocated at L1; if a star overflows its Roche lobe, thenit will lose matter to its companion star.
19.4 Mathematical details
Lagrangian points are the constant-pattern solutions ofthe restricted three-body problem. For example, giventwo massive bodies in orbits around their commonbarycenter, there are five positions in space where a thirdbody, of comparatively negligible mass, could be placedso as to maintain its position relative to the two mas-sive bodies. As seen in a rotating reference frame that
L1L2
L3
L4
L5
A contour plot of the effective potential due to gravity and thecentrifugal force of a two-body system in a rotating frame of ref-erence. The arrows indicate the gradients of the potential aroundthe five Lagrange points—downhill toward them (red) or awayfrom them (blue). Counterintuitively, the L4 and L5 points arethe high points of the potential. At the points themselves theseforces are balanced.
Visualisation of the relationship between the Lagrangian points(red) of a planet (blue) orbiting a star (yellow) anticlockwise,and the effective potential in the plane containing the orbit (greyrubber-sheet model with purple contours of equal potential).[10]
Click for animation.
matches the angular velocity of the two co-orbiting bod-ies, the gravitational fields of two massive bodies com-bined with the minor body’s centrifugal force are in bal-ance at the Lagrangian points, allowing the smaller thirdbody to be relatively stationary with respect to the firsttwo.[11]
19.4.1 L1
The location of L1 is the solution to the following equa-tion, balancing gravitation and the centrifugal force:
M1
(R−r)2 = M2
r2 + M1
R2 − r(M1+M2)R3
where r is the distance of the L1 point from the smallerobject, R is the distance between the two main objects,
19.5. STABILITY 79
and M1 and M2 are the masses of the large and smallobject, respectively. (The quantity in parentheses on theright is the distance of L1 from the center of mass.) Solv-ing this for r involves solving a quintic function, but ifthe mass of the smaller object (M2) is much smaller thanthe mass of the larger object (M1) then L1 and L2 are atapproximately equal distances r from the smaller object,equal to the radius of the Hill sphere, given by:
r ≈ R 3
√M2
3M1
This distance can be described as being such that theorbital period, corresponding to a circular orbit with thisdistance as radius aroundM2 in the absence ofM1, is thatof M2 around M1, divided by
√3 ≈ 1.73 :
Ts,M2(r) =TM2,M1(R)√
3.
19.4.2 L2
The location of L2 is the solution to the following equa-tion, balancing gravitation and inertia:
M1
(R+r)2 + M2
r2 = M1
R2 + r(M1+M2)R3
with parameters defined as for the L1 case. Again, if themass of the smaller object (M2) is much smaller than themass of the larger object (M1) then L2 is at approximatelythe radius of the Hill sphere, given by:
r ≈ R 3
√M2
3M1
19.4.3 L3
The location of L3 is the solution to the following equa-tion, balancing gravitation and the centrifugal force:
M1
(R−r)2 + M2
(2R−r)2 =(
M2
M1+M2R+R− r
)M1+M2
R3
with parameters defined as for the L1 and L2 cases exceptthat r now indicates how much closer L3 is to the moremassive object than the smaller object. If the mass of thesmaller object (M2) is much smaller than the mass of thelarger object (M1) then:
r ≈ R7M2
12M1
19.4.4 L4 and L5
Further information: Trojan (astronomy)
The reason these points are in balance is that, at L4 andL5, the distances to the two masses are equal. Accord-ingly, the gravitational forces from the two massive bod-ies are in the same ratio as the masses of the two bod-ies, and so the resultant force acts through the barycenterof the system; additionally, the geometry of the triangleensures that the resultant acceleration is to the distancefrom the barycenter in the same ratio as for the two mas-sive bodies. The barycenter being both the center of massand center of rotation of the three-body system, this re-sultant force is exactly that required to keep the smallerbody at the Lagrange point in orbital equilibrium with theother two larger bodies of system. (Indeed, the third bodyneed not have negligible mass.) The general triangularconfiguration was discovered by Lagrange in work on thethree-body problem.
19.5 Stability
Although the L1, L2, and L3 points are nominally unsta-ble, it turns out that it is possible to find (unstable) peri-odic orbits around these points, at least in the restrictedthree-body problem. These periodic orbits, referred to as“halo” orbits, do not exist in a full n-body dynamical sys-tem such as the Solar System. However, quasi-periodic(i.e. bounded but not precisely repeating) orbits follow-ing Lissajous-curve trajectories do exist in the n-bodysystem. These quasi-periodic Lissajous orbits are whatmost of Lagrangian-point missions to date have used. Al-though they are not perfectly stable, a relatively modesteffort at station keeping can allow a spacecraft to stay ina desired Lissajous orbit for an extended period of time.It also turns out that, at least in the case of Sun–Earth-L1
missions, it is actually preferable to place the spacecraftin a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit, instead of having it sit atthe Lagrangian point, because this keeps the spacecraftoff the direct line between Sun and Earth, thereby reduc-ing the impact of solar interference on Earth–spacecraftcommunications. Similarly, a large-amplitude Lissajousorbit around L2 can keep a probe out of Earth’s shadowand therefore ensures a better illumination of its solarpanels.
19.6 Spaceflight applications
Earth–Moon L1 allows comparatively easy access to Lu-nar and Earth orbits with minimal change in velocity andhas this as an advantage to position a half-way mannedspace station intended to help transport cargo and per-sonnel to the Moon and back.Earth–Moon L2 would be a good location for acommunications satellite covering the Moon’s far sideand would be “an ideal location” for a propellant depotas part of the proposed depot-based space transportation
80 CHAPTER 19. LAGRANGIAN POINT
architecture.[12]
The satellite ACE in an orbit around L1
Sun–Earth L1 is suited for making observations of theSun–Earth system. Objects here are never shadowed byEarth or the Moon. The first mission of this type wasthe International Sun Earth Explorer 3 (ISEE-3) missionused as an interplanetary early warning stormmonitor forsolar disturbances.Sun–Earth L2 is a good spot for space-based observa-tories. Because an object around L2 will maintain thesame relative position with respect to the Sun and Earth,shielding and calibration are much simpler. It is, how-ever, slightly beyond the reach of Earth’s umbra,[13] sosolar radiation is not completely blocked. From thispoint, the Sun, Earth and Moon are relatively closely po-sitioned together in the sky, and hence leave a large fieldof view without interference – this is especially helpfulfor infrared astronomy.Sun–Earth L3 was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Oncespace-based observation became possible via satellites[14]and probes, it was shown to hold no such object. TheSun–Earth L3 is unstable and could not contain an ob-ject, large or small, for very long. This is because thegravitational forces of the other planets are stronger thanthat of Earth (Venus, for example, comes within 0.3 AUof this L3 every 20 months).A spacecraft orbiting near Sun–Earth L3 would be ableto closely monitor the evolution of active sunspot regionsbefore they rotate into a geoeffective position, so that a7-day early warning could be issued by the NOAA SpaceWeather Prediction Center. Moreover, a satellite nearSun–Earth L3 would provide very important observationsnot only for Earth forecasts, but also for deep space sup-port (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectoriesto Sun–Earth L3 were studied and several designs wereconsidered.[15]
Scientists at the B612 Foundation are planning to useVenus's L3 point to position their planned Sentinel tele-scope, which aims to look back towards Earth’s orbit andcompile a catalogue of near-Earth asteroids.[16]
Missions to Lagrangian points generally orbit the pointsrather than occupy them directly.
Another interesting and useful property of the collinearLagrangian points and their associated Lissajous orbits isthat they serve as “gateways” to control the chaotic tra-jectories of the Interplanetary Transport Network.
19.6.1 Spacecraft at Sun–Earth L1
International Sun Earth Explorer 3 (ISEE-3) began itsmission at the Sun–Earth L1 before leaving to intercepta comet in 1982. The Sun–Earth L1 is also the point towhich the Reboot ISEE-3 mission was attempting to re-turn the craft as the first phase of a recovery mission (asof September 25, 2014 all efforts have failed and contactwas lost).[17]
Solar and Heliospheric Observatory (SOHO) is stationedin a halo orbit at L1, and the Advanced Composition Ex-plorer (ACE) in a Lissajous orbit. WIND is also at L1.Deep Space Climate Observatory (DSCOVR), launchedon 11 February 2015, began orbiting L1 on 8 June 2015to study the solar wind and its effects on Earth.[18]
19.6.2 Spacecraft at Sun–Earth L2
Spacecraft at the Sun–Earth L2 point are in a Lissajousorbit until decommissioned, when they are sent into aheliocentric graveyard orbit.
• 1 October 2001 – October 2010—Wilkinson Mi-crowave Anisotropy Probe[19]
• July 2009 – 29 April 2013—Herschel Space Obser-vatory[20]
• 3 July 2009 – 21 October 2013—Space observatoryPlanck
• 25 August 2011 – April 2012—Chang'e 2,[21][22]from where it travelled to 4179 Toutatis and theninto deep space
• January 2014 – 2018—Gaia probe
• 2018—JamesWebb Space Telescope will use a haloorbit
• 2020—Euclid observatory
• 2028—Advanced Telescope for High Energy Astro-physics will use a halo orbit
19.6.3 List of missions to Lagrangianpoints
Color key:
19.9. REFERENCES 81
Past and present missions
Future and proposed missions
19.7 See also• Co-orbital configuration
• Euler’s three-body problem
• Gegenschein
• Hill sphere
• Horseshoe orbit
• L5 Society
• Lagrange point colonization
• Lagrangian mechanics
• Lissajous orbit
• List of objects at Lagrangian points
• Lunar space elevator
• Trojan wave packets
19.8 Notes[1] Actually 25+
√621
2≈ 24.9599357944
19.9 References[1] Koon, W. S.; M.W. Lo; J. E. Marsden; S. D. Ross (2006).
Dynamical Systems, the Three-Body Problem, and SpaceMission Design. p. 9. (16MB)
[2] Leonhard Euler, De motu rectilineo trium corporum semutuo attrahentium (1765)
[3] Lagrange, Joseph-Louis (1867–92). “Tome 6, ChapitreII: Essai sur le problème des trois corps”. Oeuvres de La-grange (in French). Gauthier-Villars. pp. 229–334.
[4] Cornish, Neil J. “The Lagrangian Points” (PDF). Depart-ment of Physics, Bozeman Campus, Montana State Uni-versity, USA. Retrieved 29 July 2011.
[5] The Lagrange Points PDF, Neil J. Cornish with input fromJeremy Goodman
[6] Space.com: First Asteroid Companion of Earth Discoveredat Last
[7] NASA—NASA’s Wise Mission Finds First Trojan Aster-oid Sharing Earth’s Orbit
[8] “A Search for Natural or Artificial Objects Located atthe Earth–Moon Libration Points” by Robert Freitas andFrancisco Valdes, Icarus 42, 442-447 (1980)
[9] “List Of Neptune Trojans”. Minor Planet Center.Archived from the original on 2011-08-23. Retrieved2010-10-27.
[10] ZF Seidov, “The Roche Problem: Some Analytics”, TheAstrophysical Journal, 603:283-284, 2004 March 1
[11] "Lagrange Points" by Enrique Zeleny, Wolfram Demon-strations Project.
[12] Zegler, Frank; Bernard Kutter (2010-09-02). “Evolvingto a Depot-Based Space Transportation Architecture”(PDF). AIAA SPACE 2010 Conference & Exposition.AIAA. p. 4. Retrieved 2011-01-25. L2 is in deep spacefar away from any planetary surface and hence the ther-mal, micrometeoroid, and atomic oxygen environmentsare vastly superior to those in LEO. Thermodynamic stasisand extended hardware life are far easier to obtain with-out these punishing conditions seen in LEO. L2 is not justa great gateway—it is a great place to store propellants.... L2 is an ideal location to store propellants and car-gos: it is close, high energy, and cold. More importantly,it allows the continuous onward movement of propellantsfrom LEO depots, thus suppressing their size and effec-tively minimizing the near-Earth boiloff penalties.
[13] Angular size of the Sun at 1 AU + 930000 miles: 31.6',angular size of Earth at 930000 miles: 29.3'
[14] STEREO mission description by NASA,http://www.nasa.gov/mission_pages/stereo/main/index.html#.UuG0NxDb-kk
[15] Tantardini, Marco; Fantino, Elena; Yuan Ren; PierpaoloPergola; Gerard Gómez; Josep J. Masdemont (2010).“Spacecraft trajectories to the L3 point of the Sun–Earththree-body problem”. Celestial Mechanics and DynamicalAstronomy (Springer).
[16] The Sentinel Mission, B612 Foundation. Retrieved Feb2014.
[17] “ISEE-3 is in Safe Mode”. Space College. 25 September2014. “The ground stations listening to ISEE-3 have notbeen able to obtain a signal since Tuesday the 16th”
[18] http://www.nesdis.noaa.gov/news_archives/DSCOVR_L1_orbit.html
[19] “Mission Complete! WMAP Fires Its Thrusters For TheLast Time”.
[20] Toobin, Adam (2013-06-19). “Herschel Space TelescopeShut Down For Good, ESA Announces”. Huffington Post.
[21] Lakdawalla, Emily (14 June 2012). “Chang'E 2 has de-parted Earth’s neighborhood for.....asteroid Toutatis!?".Retrieved 15 June 2012.
[22] Lakdawalla, Emily (15 June 2012). “Update on yester-day’s post about Chang'E 2 going to Toutatis”. PlanetarySociety. Retrieved 26 June 2012.
[23] “Solar System Exploration: ISEE-3/ICE”. NASA. Re-trieved 2010-09-28.
[24] Lakdawalla, Emily (October 3, 2008). “It’s Alive!". ThePlanetary Science Weblog.
82 CHAPTER 19. LAGRANGIAN POINT
[25] Chang, Kenneth (August 8, 2014). “Rudderless Craftto Get Glimpse of Home Before Sinking Into Space’sDepths”. The New York Times.
[26] “ACEMission”. Caltech ACE Science Center. Retrieved2013-03-18.
[27] Sullivan, Brian K (12 June 2015). “Space 'Buoy' MayHelp Protect Power Grid From Sun’s Fury”. Bloomberg.
[28] “SOHO’s Orbit: An Uninterrupted View of the Sun”.NASA. Retrieved 2010-09-28.
[29] “WIND Spacecraft”. NASA. Retrieved 2010-09-28.
[30] “WMAP Facts”. NASA. Retrieved 2013-03-18.
[31] http://map.gsfc.nasa.gov/news/events.html WMAPCeases Communications
[32] “Herschel Factsheet”. European Space Agency. 17 April2009. Retrieved 2009-05-12.
[33] “Herschel space telescope finishes mission”. BBC news.29 April 2013.
[34] “Last command sent to ESA’s Planck space telescope”.European Space Agency. October 23, 2013. RetrievedOctober 23, 2013.
[35] Fox, Karen C. “First ARTEMIS Spacecraft SuccessfullyEnters Lunar Orbit”. The Sun-Earth Connection: Helio-physics. NASA.
[36] Hendrix, Susan. “Second ARTEMIS Spacecraft Success-fully Enters Lunar Orbit”. The Sun-Earth Connection: He-liophysics. NASA.
[37] “Worldwide launch schedule”. Spaceflight Now. 27November 2013.
[38] “ESA Gaia home”. ESA. Retrieved 23 October 2013.
[39] P. E. Schmid (June 1968). “Lunar Far-Side Communica-tion Satellites” (PDF). NASA. Retrieved 2008-07-16.
[40] O'Neill, Gerard K. (September 1974). “The Coloniza-tion of Space”. Physics Today (American Institute ofPhysics) 27 (9): 32–40. Bibcode:1974PhT....27i..32O.doi:10.1063/1.3128863.
[41] “LISA Pathfinder factsheet”. ESA. 11 June 2012. Re-trieved 26 June 2012.
[42] http://www.business-standard.com/article/beyond-business/man-in-space-and-other-plans-114111401887_1.html
[43] “JWST factsheet”. ESA. 2013-09-04. Retrieved 2013-09-07.
[44] “NASA Officially Joins ESA’s 'Dark Universe' Mission”.JPL/NASA. 24 January 2013. Retrieved 12 April 2013.
[45] Paul Hertz (2013-06-04), NASA Astrophysics presentationto American Astronomical Society (PDF), retrieved 2013-09-10
[46] Hiroshi Shibai (2014-12-31), SPICA (PDF), retrieved2015-02-24
[47] NASA teams evaluating ISS-built Exploration Platformroadmap
[48] Bergin, Chris (December 2011). “Exploration Gate-way Platform hosting Reusable Lunar Lander proposed”.NASA Spaceflight.com. Retrieved 2011-12-05.
[49] “ESA Science & Technology: Athena to study the hot andenergetic Universe”. ESA. 27 June 2014. Retrieved 23August 2014.
19.10 External links• Joseph-Louis, Comte Lagrange, from OeuvresTome 6, “Essai sur le Problème des Trois Corps”—Essai (PDF); source Tome 6 (Viewer)
• “Essay on the Three-Body Problem” by J-L La-grange, translated from the above, in http://www.merlyn.demon.co.uk/essai-3c.htm.
• Considerationes de motu corporum coelestium—Leonhard Euler—transcription and translation athttp://www.merlyn.demon.co.uk/euler304.htm.
• What are Lagrange points?—European SpaceAgency page, with good animations
• Explanation of Lagrange points—Prof. Neil J. Cor-nish
• A NASA explanation—also attributed to Neil J.Cornish
• Explanation of Lagrange points—Prof. John Baez
• Geometry and calculations of Lagrange points—DrJ R Stockton
• Locations of Lagrange points, with approxima-tions—Dr. David Peter Stern
• An online calculator to compute the precise posi-tions of the 5 Lagrange points for any 2-body sys-tem—Tony Dunn
• Astronomy cast—Ep. 76: Lagrange Points FraserCain and Dr. Pamela Gay
• The Five Points of Lagrange by Neil deGrasseTyson
• Earth, a lone Trojan discovered
Chapter 20
Lagrangian system
In mathematics, a Lagrangian system is a pair (Y,L), consisting of a smooth fiber bundle Y → X and aLagrangian density L, which yields the Euler–Lagrangedifferential operator acting on sections of Y → X.
In classical mechanics, many dynamical systems are La-grangian systems. The configuration space of such a La-grangian system is a fiber bundle Q → ℝ over the timeaxis ℝ. In particular, Q = ℝ × M if a reference frameis fixed. In classical field theory, all field systems are theLagrangian ones.
20.1 Lagrangians and Euler–Lagrange operators
A Lagrangian density L (or, simply, a Lagrangian) oforder r is defined as an n-form, n = dim X, on the r-orderjet manifold JrY of Y.A Lagrangian L can be introduced as an element of thevariational bicomplex of the differential graded algebraO∗∞(Y) of exterior forms on jet manifolds of Y → X.The coboundary operator of this bicomplex contains thevariational operator δ which, acting on L, defines the as-sociated Euler–Lagrange operator δL.
20.1.1 In coordinates
Given bundle coordinates xλ, yi on a fiber bundle Y andthe adapted coordinates xλ, yi, yiΛ, (Λ = (λ1, ...,λk), |Λ| =k ≤ r) on jet manifolds JrY, a Lagrangian L and its Euler–Lagrange operator read
L = L(xλ, yi, yiΛ) dnx,
δL = δiL dyi∧dnx, δiL = ∂iL+∑|Λ|
(−1)|Λ| dΛ ∂Λi L,
where
dΛ = dλ1 · · · dλk, dλ = ∂λ + yiλ∂i + · · · ,
denote the total derivatives.For instance, a first-order Lagrangian and its second-order Euler–Lagrange operator take the form
L = L(xλ, yi, yiλ) dnx, δiL = ∂iL − dλ∂
λi L.
20.1.2 Euler–Lagrange equations
The kernel of an Euler–Lagrange operator provides theEuler–Lagrange equations δL = 0.
20.2 Cohomology and Noether’stheorems
Cohomology of the variational bicomplex leads to the so-called variational formula
dL = δL+ dHΘL,
where
dHϕ = dxλ ∧ dλϕ, ϕ ∈ O∗∞(Y )
is the total differential and θL is a Lepage equivalent ofL. Noether’s first theorem and Noether’s second theoremare corollaries of this variational formula.
20.3 Graded manifolds
Extended to graded manifolds, the variational bicomplexprovides description of graded Lagrangian systems ofeven and odd variables.[1]
20.4 Alternative formulations
In a different way, Lagrangians, Euler–Lagrange opera-tors and Euler–Lagrange equations are introduced in the
83
84 CHAPTER 20. LAGRANGIAN SYSTEM
framework of the calculus of variations.
20.5 Classical mechanics
In classical mechanics equations of motion are first andsecond order differential equations on a manifold M orvarious fiber bundles Q over ℝ. A solution of the equa-tions of motion is called a motion.[2][3]
20.6 See also• Lagrangian mechanics
• Calculus of variations
• Noether’s theorem
• Noether identities
• Jet bundle
• Jet (mathematics)
• Variational bicomplex
20.7 References[1] Sardanashvily 2013
[2] Arnold 1989, p. 83
[3] Giachetta, Mangiarotti & Sardanashvily 2011, p. 7
• Arnold, V. I. (1989),Mathematical Methods of Clas-sical Mechanics, Graduate Texts in Mathematics 60(second ed.), Springer-Verlag, ISBN 0-387-96890-3
• Giachetta, G.; Mangiarotti, L.; Sardanashvily, G.(1997). New Lagrangian and Hamiltonian Methodsin Field Theory. World Scientific. ISBN 981-02-1587-8.
• Giachetta, G.; Mangiarotti, L.; Sardanashvily, G.(2011). Geometric formulation of classical andquantum mechanics. World Scientific. ISBN 978-981-4313-72-8.
• Olver, P. (1993). Applications of Lie Groups toDifferential Equations (2ed ed.). Springer-Verlag.ISBN 0-387-94007-3.
• Sardanashvily, G. (2013). “Graded Lagrangianformalism”. Int. G. Geom. Methods Mod.Phys. (World Scientific) 10 (5). arXiv:1206.2508.doi:10.1142/S0219887813500163. ISSN 0219-8878.
20.8 External links• Sardanashvily, G. (2009). “Fibre Bundles, Jet Man-ifolds and Lagrangian Theory. Lectures for Theo-reticians”. arXiv:0908.1886.
Chapter 21
Minimal coupling
In physics, minimal coupling refers to a coupling be-tween fields which involves only the charge distributionand not higher multipole moments of the charge distribu-tion. This minimal coupling is in contrast to, for example,Pauli coupling, which includes the magnetic moment ofan electron directly in the Lagrangian.In electrodynamics, minimal coupling is adequate to ac-count for all electromagnetic interactions. Higher mo-ments of particles are consequences of minimal couplingand non-zero spin.Mathematically, minimal coupling is achieved by sub-tracting the charge ( q ) times the four-potential ( Aµ
) from the four-momentum ( pµ ) in the Lagrangian orHamiltonian:
pµ 7→ pµ − q Aµ
Taken almost verbatim from Doughty’s La-grangian Interaction, pg. 456[1]
See the Hamiltonian mechanics article for a full deriva-tion and examples.
21.1 Inflation
In studies of cosmological inflation, minimal coupling ofa scalar field usually refers to minimal coupling to grav-ity. This means that the action for the field φ (calledthe inflaton in the context of inflation) is not coupled tothe scalar curvature. Its only coupling to gravity is thecoupling to the Lorentz invariant measure √
g d4x con-structed from the metric (in Planck units):
S =
∫d4x
√g
(−1
2R+
1
2∇µφ∇µφ− V (φ)
)where g := det gµν .
21.2 See also• Gauge covariant derivative
• Hamiltonian mechanics
• Lagrangian mechanics
21.3 References[1] Doughty, Noel (1990). Lagrangian Interaction. Westview
Press. ISBN 0-201-41625-5.
85
Chapter 22
Monogenic system
In classical mechanics, a physical system is termed amonogenic system if the force acting on the system canbe modelled in an especially convenient mathematicalform (see mathematical definition below). In physics,among the most studied physical systems are monogenicsystems.In Lagrangian mechanics, the property of being mono-genic is a necessary condition for the equivalence ofdifferent formulations of principle. If a physical sys-tem is both a holonomic system and a monogenic sys-tem, then it is possible to derive Lagrange’s equationsfrom d'Alembert’s principle; it is also possible to deriveLagrange’s equations from Hamilton’s principle.[1]
The term was introduced by Cornelius Lanczos inhis book The Variational Principles of Mechanics(1970).[2][3]
Monogenic systems have excellent mathematical charac-teristics and are well suited for mathematical analysis.Pedagogically, within the discipline of mechanics, it isconsidered a logical starting point for any serious physicsendeavour.
22.1 Mathematical definition
In a physical system, if all forces, with the exception ofthe constraint forces, are derivable from the generalizedscalar potential, and this generalized scalar potential is afunction of generalized coordinates, generalized veloci-ties, or time, then, this system is amonogenic system.Expressed using equations, the exact relationship be-tween generalized force Fi and generalized potentialV(q1, q2, . . . , qN , q1, q2, . . . , qN , t) is as follows:
Fi = − ∂V∂qi
+d
dt
(∂V∂qi
);
where qi is generalized coordinate, qi is generalized ve-locity, and t is time.If the generalized potential in a monogenic system de-pends only on generalized coordinates, and not on gen-eralized velocities and time, then, this system is a
conservative system.The relationship between general-ized force and generalized potential is as follows:
Fi = − ∂V∂qi
22.2 See also• Lagrangian mechanics
• Hamiltonian mechanics
• Holonomic system
• Scleronomous
22.3 References[1] Goldstein, Herbert; Poole, Charles P., Jr.; Safko, John
L. (2002). Classical Mechanics (3rd ed.). San Fran-cisco, CA: Addison Wesley. pp. 18–21,45. ISBN 0-201-65702-3.
[2] J., Butterfield (3 September 2004). “Between Laws andModels: Some Philosophical Morals of Lagrangian Me-chanics”. PhilSci-Archive. p. 43. Retrieved 23 January2015.
[3] Cornelius, Lanczos (1970). The Variational Principles ofMechanics. Toronto: University of Toronto Press. p. 30.ISBN 0-8020-1743-6.
86
Chapter 23
Ostrogradsky instability
In applied mathematics, theOstrogradsky instability isa consequence of a theorem of Mikhail Ostrogradsky inclassical mechanics according to which a non-degenerateLagrangian dependent on time derivatives of higher thanthe first corresponds to a linearly unstable Hamiltonianassociated with the Lagrangian via a Legendre transform.The Ostrogradsky instability has been proposed as an ex-planation as to why no differential equations of higher or-der than two appear to describe physical phenomena.[1]
23.1 Outline of proof [2]
The main points of the proof can be made clearer byconsidering a one-dimensional system with a LagrangianL(q, q, q) . The Euler-Lagrange equation is
dL
dq− d
dt
dL
dq+
d2
dt2dL
dq= 0.
Non-degeneracy of L means that the canonical coordi-nates can be expressed in terms of the derivatives ofq and vice versa. Thus, dL/dq is a function of q (ifit was not, the Jacobian det[d2L/(dqi dqj)] would van-ish, which would mean that L is degenerate), meaningthat we can write q(4) = F (q, q, q, q(3)) or, inverting,q = G(t, q0, q0, q0, q
(3)0 ) . Since the evolution of q de-
pends upon four initial parameters, this means that thereare four canonical coordinates. We can write those as
Q1 := q
Q2 := q
and by using the definition of the conjugate momentum,
P1 :=dL
dq− d
dt
dL
dq
P2 :=dL
dq
Due to non-degeneracy, we can write q as q =a(Q1, Q2, P2) . Note that only three arguments are
needed since the Lagrangian itself only has three free pa-rameters. By Legendre transforming, we find the Hamil-tonian to be
H = P1Q2 − P2a(Q1, Q2, P2)
We now notice that the Hamiltonian is linear in P1 . Thisis Ostrogradsky’s instability, and it stems from the factthat the Lagrangian depends on fewer coordinates thanthere are canonical coordinates (which correspond to theinitial parameters needed to specify the problem). Theextension to higher dimensional systems is analogous, andthe extension to higher derivatives simply mean that thephase space is of even higher dimension than the con-figuration space, which exacerbates the instability (sincethe Hamiltonian is linear in even more canonical coordi-nates).
23.2 Notes[1] HayatoMotohashi, Teruaki Suyama (2014). “Third-order
equations of motion and the Ostrogradsky instability”.
[2] R. P. Woodard (2006). “Avoiding Dark Energy with 1/RModifications of Gravity”.
87
Chapter 24
Palatini variation
In general relativity and gravitation the Palatini varia-tion is nowadays thought of as a variation of a Lagrangianwith respect to the connection.In fact, as is well known, the Einstein–Hilbert action forgeneral relativity was first formulated purely in terms ofthe spacetime metric gµν . In the Palatini variationalmethod one takes as independent field variables not onlythe ten components gµν but also the forty components ofthe affine connection Γα
βµ , assuming, a priori, no depen-dence of the Γα
βµ from the gµν and their derivatives.The reason the Palatini variation is considered impor-tant is that it means that the use of the Christoffel con-nection in general relativity does not have to be addedas a separate assumption; the information is already inthe Lagrangian. For theories of gravitation which havemore complex Lagrangians than the Einstein–Hilbert La-grangian of general relativity, the Palatini variation some-times gives more complex connections and sometimestensorial equations.Attilio Palatini (1889–1949) was an Italian mathemati-cian who received his doctorate from the University ofPadova, where he studied under Levi-Civita and Ricci-Curbastro.The history of the subject, and Palatini’s connection withit, are not straightforward (see references). In fact, itseems that what the textbooks now call “Palatini formal-ism” was actually invented in 1925 by Einstein, and as theyears passed, people tended to mix up the Palatini iden-tity and the Palatini formalism.
24.1 See also• Palatini identity
• Self-dual Palatini action
• Tetradic Palatini action
24.2 References• Palatini, A. (1919). “Deduzione invariantiva delleequazioni gravitazionali dal principio di Hamilton”.
Rend. Circ. Mat. Palermo 43: 203–212. [Englishtranslation by R. Hojman and C. Mukku in P. G.Bergmann and V. De Sabbata (eds.) Cosmology andGravitation, Plenum Press, New York (1980)]
• Tsamparlis, M. (1978). “On the Palatini methodof variation”. J. Math. Phys. 19 (3): 555.doi:10.1063/1.523699.
• Ferraris, M.; Francaviglia, M.; Reina, C. (1982).“Variational Formulation of General Relativity from1915 to 1925 'Palatini’s Method' Discovered by Ein-stein in 1925”. Gen. Rel. Grav. 14: 243–254.
88
Chapter 25
Rayleigh dissipation function
In physics, the Rayleigh dissipation function, namedfor Lord Rayleigh, is a function used to handle the effectsof velocity-proportional frictional forces in Lagrangianmechanics. It is defined for a system of N particles as
F =1
2
N∑i=1
(kxv2i,x + kyv
2i,y + kzv
2i,z).
The force of friction is negative the velocity gradient ofthe dissipation function, Ff = −∇vF . The function ishalf the rate at which energy is being dissipated by thesystem through friction.
25.1 References• Goldstein, Herbert (1980). Classical Mechanics(2nd ed.). Reading, MA: Addison-Wesley. p. 24.ISBN 0-201-02918-9.
89
Chapter 26
Rheonomous
A mechanical system is rheonomous if its equations ofconstraints contain the time as an explicit variable.[1][2]Such constraints are called rheonomic constraints. Theopposite of rheonomous is scleronomous.[1][2]
26.1 Example: simple 2D pendu-lum
A simple pendulum
As shown at right, a simple pendulum is a system com-posed of a weight and a string. The string is attachedat the top end to a pivot and at the bottom end to aweight. Being inextensible, the string has a constantlength. Therefore this system is scleronomous; it obeysthe scleronomic constraint
√x2 + y2 − L = 0
where (x, y) is the position of the weight andL the lengthof the string.The situation changes if the pivot point is moving, e.g.undergoing a simple harmonic motion
A simple pendulum with oscillating pivot point
xt = x0 cosωt
where x0 is the amplitude, ω the angular frequency, andt time.Although the top end of the string is not fixed, the lengthof this inextensible string is still a constant. The distancebetween the top end and the weight must stay the same.Therefore this system is rheonomous; it obeys the rheo-nomic constraint
√(x− x0 cosωt)2 + y2 − L = 0
26.2 See also
• Lagrangian mechanics
• Holonomic constraints
90
26.3. REFERENCES 91
26.3 References[1] Goldstein, Herbert (1980). Classical Mechanics (2nd ed.).
United States of America: Addison Wesley. p. 12. ISBN0-201-02918-9. Constraints are further classified accord-ing as the equations of constraint contain the time as anexplicit variable (rheonomous) or are not explicitly depen-dent on time (scleronomous).
[2] Spiegel, Murray R. (1994). Theory and Problems ofTHEORETICAL MECHANICS with an Introduction to La-grange’s Equations and Hamiltonian Theory. Schaum’sOutline Series. McGraw Hill. p. 283. ISBN 0-07-060232-8. In many mechanical systems of importancethe time t does not enter explicitly in the equations (2) or(3). Such systems are sometimes called scleronomic. Inothers, as for example those involving moving constraints,the time t does enter explicitly. Such systems are calledrheonomic.
Chapter 27
Scleronomous
Amechanical system is scleronomous if the equations ofconstraints do not contain the time as an explicit variable.Such constraints are called scleronomic constraints.
27.1 Application
Main article:Generalized velocity
In 3-D space, a particle with mass m , velocity v haskinetic energy
T =1
2mv2.
Velocity is the derivative of position with respect time.Use chain rule for several variables:
v = drdt
=∑i
∂r∂qi
qi +∂r∂t
.
Therefore,
T =1
2m
(∑i
∂r∂qi
qi +∂r∂t
)2
.
Rearranging the terms carefully,[1]
T = T0 + T1 + T2 :
T0 =1
2m
(∂r∂t
)2
,
T1 =∑i
m∂r∂t
· ∂r∂qi
qi,
T2 =∑i,j
1
2m
∂r∂qi
· ∂r∂qj
qiqj ,
where T0 , T1 , T2 are respectively homogeneous func-tions of degree 0, 1, and 2 in generalized velocities. If
this system is scleronomous, then the position does notdepend explicitly with time:
∂r∂t
= 0.
Therefore, only term T2 does not vanish:
T = T2.
Kinetic energy is a homogeneous function of degree 2 ingeneralized velocities .
27.2 Example: pendulum
A simple pendulum
As shown at right, a simple pendulum is a system com-posed of a weight and a string. The string is attachedat the top end to a pivot and at the bottom end to aweight. Being inextensible, the string’s length is a con-stant. Therefore, this system is scleronomous; it obeysscleronomic constraint
92
27.4. REFERENCES 93
√x2 + y2 − L = 0,
where (x, y) is the position of the weight and L is lengthof the string.
A simple pendulum with oscillating pivot point
Take a more complicated example. Refer to the next fig-ure at right, Assume the top end of the string is attachedto a pivot point undergoing a simple harmonic motion
xt = x0 cosωt,
where x0 is amplitude, ω is angular frequency, and t istime.Although the top end of the string is not fixed, the lengthof this inextensible string is still a constant. The distancebetween the top end and the weight must stay the same.Therefore, this system is rheonomous as it obeys con-straint explicitly dependent on time
√(x− x0 cosωt)2 + y2 − L = 0.
27.3 See also
• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous
27.4 References[1] Goldstein, Herbert (1980). Classical Mechanics (3rd ed.).
United States of America: Addison Wesley. p. 25. ISBN0-201-65702-3.
Chapter 28
Tautological one-form
In mathematics, the tautological one-form is a spe-cial 1-form defined on the cotangent bundle T*Q of amanifold Q. The exterior derivative of this form defines asymplectic form giving T*Q the structure of a symplecticmanifold. The tautological one-form plays an importantrole in relating the formalism of Hamiltonian mechan-ics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form,the Poincaré one-form, the canonical one-form, or thesymplectic potential. A similar object is the canonicalvector field on the tangent bundle. In algebraic geometryand complex geometry the term “canonical” is discour-aged, due to confusion with the canonical class, and theterm “tautological” is preferred, as in tautological bundle.In canonical coordinates, the tautological one-form isgiven by
θ =∑i
pidqi
Equivalently, any coordinates on phase space which pre-serve this structure for the canonical one-form, up to atotal differential (exact form), may be called canonicalcoordinates; transformations between different canonicalcoordinate systems are known as canonical transforma-tions.The canonical symplectic form, also known as thePoincaré two-form, is given by
ω = −dθ =∑i
dqi ∧ dpi
The extension of this concept to general fibre bundles isknown as the solder form.
28.1 Coordinate-free definition
The tautological 1-form can also be defined rather ab-stractly as a form on phase space. Let Q be a manifoldand M = T ∗Q be the cotangent bundle or phase space.Let
π : M → Q
be the canonical fiber bundle projection, and let
Tπ : TM → TQ
be the induced tangent map. Letm be a point onM. SinceM is the cotangent bundle, we can understand m to be amap of the tangent space at q = π(m) :
m : TqQ → R
That is, we have that m is in the fiber of q. The tautolog-ical one-form θm at point m is then defined to be
θm = m Tπ
It is a linear map
θm : TmM → R
and so
θ : M → T ∗M
28.2 Properties
The tautological one-form is the unique horizontal one-form that “cancels” a pullback. That is, let
β : Q → T ∗Q
be any 1-form on Q, and (considering it as a map from Qto T*Q ) let β∗ be its pullback. Then
β∗θ = β
94
28.5. SEE ALSO 95
which can be most easily understood in terms of coordi-nates:
β∗θ = β∗(∑i
pi dqi) =
∑i
β∗pi dqi =
∑i
βi dqi = β.
So, by the commutation between the pull-back and theexterior derivative,
β∗ω = −β∗dθ = −d(β∗θ) = −dβ
28.3 Action
If H is a Hamiltonian on the cotangent bundle andXH isits Hamiltonian flow, then the corresponding action S isgiven by
S = θ(XH)
In more prosaic terms, the Hamiltonian flow representsthe classical trajectory of a mechanical system obeyingthe Hamilton-Jacobi equations of motion. The Hamilto-nian flow is the integral of the Hamiltonian vector field,and so one writes, using traditional notation for action-angle variables:
S(E) =∑i
∮pi dq
i
with the integral understood to be taken over themanifolddefined by holding the energy E constant: H = E =const. .
28.4 On metric spaces
If the manifold Q has a Riemannian or pseudo-Riemannian metric g, then corresponding definitions canbe made in terms of generalized coordinates. Specifi-cally, if we take the metric to be a map
g : TQ → T ∗Q
then define
Θ = g∗θ
and
Ω = −dΘ = g∗ω
In generalized coordinates (q1, . . . , qn, q1, . . . , qn) onTQ, one has
Θ =∑ij
gij qidqj
and
Ω =∑ij
gij dqi ∧ dqj +
∑ijk
∂gij∂qk
qi dqj ∧ dqk
The metric allows one to define a unit-radius sphere inT ∗Q . The canonical one-form restricted to this sphereforms a contact structure; the contact structure may beused to generate the geodesic flow for this metric.
28.5 See also• fundamental class
• solder form
28.6 References• Ralph Abraham and Jerrold E. Marsden, Founda-tions of Mechanics, (1978) Benjamin-Cummings,London ISBN 0-8053-0102-X See section 3.2.
Chapter 29
Total derivative
In the mathematical field of differential calculus, a totalderivative or full derivative of a function f of severalvariables, e.g., t , x , y , etc., with respect to an exogenousargument, e.g., t , is the limiting ratio of the change inthe function’s value to the change in the exogenous argu-ment’s value (for arbitrarily small changes), taking intoaccount the exogenous argument’s direct effect as well asits indirect effects via the other arguments of the func-tion.The total derivative of a function is different from its cor-responding partial derivative ( ∂ ). Calculation of the to-tal derivative of f with respect to t does not assume thatthe other arguments are constant while t varies; instead,it allows the other arguments to depend on t . The totalderivative adds in these indirect dependencies to find theoverall dependency of f on t. [1]:198-203 For example, thetotal derivative of f(t, x, y) with respect to t is
d fd t =
∂f
∂t
d td t +
∂f
∂x
dxd t +
∂f
∂y
d yd t
which simplifies to
d fd t =
∂f
∂t+
∂f
∂x
dxd t +
∂f
∂y
d yd t .
Consider multiplying both sides of the equation by thedifferential d t :
d f =∂f
∂td t+ ∂f
∂xdx+
∂f
∂yd y.
The result is the differential change d f in, or total dif-ferential of, the function f . Because f depends on t ,some of that change will be due to the partial derivativeof f with respect to t . However, some of that change willalso be due to the partial derivatives of f with respect tothe variables x and y . So, the differential d t is appliedto the total derivatives of x and y to find differentials dxand d y , which can then be used to find the contributionto d f .“Total derivative” is sometimes also used as a synonymfor the material derivative, Du
Dt , in fluid mechanics.
29.1 Differentiation with indirectdependencies
Suppose that f is a function of two variables, x and y.Normally these variables are assumed to be independent.However, in some situations they may be dependent oneach other. For example y could be a function of x, con-straining the domain of f to a curve in R2 . In this casethe partial derivative of f with respect to x does not givethe true rate of change of f with respect to changing x be-cause changing x necessarily changes y. The total deriva-tive takes such dependencies into account.For example, suppose
f(x, y) = xy
The rate of change of f with respect to x is usually thepartial derivative of f with respect to x; in this case,
∂f
∂x= y
However, if y depends on x, the partial derivative does notgive the true rate of change of f as x changes because itholds y fixed.Suppose we are constrained to the line
y = x;
then
f(x, y) = f(x, x) = x2
In that case, the total derivative of f with respect to x is
dfdx = 2x
Instead of immediately substituting for y in terms of x,this can be found equivalently using the chain rule:
96
29.3. THE TOTAL DERIVATIVE AS A LINEAR MAP 97
dfdx =
∂f
∂x+
∂f
∂y
dydx = y + x · 1 = x+ y.
Notice that this is not equal to the partial derivative:
dfdx = 2x = ∂f
∂x= y = x
While one can often perform substitutions to eliminateindirect dependencies, the chain rule provides for a moreefficient and general technique. Suppose M(t, p1, ..., pn)is a function of time t and n variables pi which themselvesdepend on time. Then, the total time derivative of M is
dMd t =
dd tM
(t, p1(t), . . . , pn(t)
).
The chain rule for differentiating a function of severalvariables implies that
dMd t =
∂M
∂t+
n∑i=1
∂M
∂pi
d pid t =
(∂
∂t+
n∑i=1
d pid t
∂
∂pi
)(M).
This expression is often used in physics for a gauge trans-formation of the Lagrangian, as two Lagrangians that dif-fer only by the total time derivative of a function of timeand the n generalized coordinates lead to the same equa-tions of motion. An interesting example concerns theresolution of causality concerning theWheeler–Feynmantime-symmetric theory. The operator in brackets (in thefinal expression) is also called the total derivative operator(with respect to t).For example, the total derivative of f(x(t), y(t)) is
d fd t =
∂f
∂x
dxd t +
∂f
∂y
d yd t .
Here there is no ∂f / ∂t term since f itself does not dependon the independent variable t directly.
29.2 The total derivative via differ-entials
Differentials provide a simple way to understand the totalderivative. For instance, suppose M(t, p1, . . . , pn) is afunction of time t and n variables pi as in the previoussection. Then, the differential of M is
dM =∂M
∂td t+
n∑i=1
∂M
∂pid pi.
This expression is often interpreted heuristically as a re-lation between infinitesimals. However, if the variables tand pi are interpreted as functions, andM(t, p1, . . . , pn)is interpreted to mean the composite of M with thesefunctions, then the above expression makes perfect senseas an equality of differential 1-forms, and is immediatefrom the chain rule for the exterior derivative. The ad-vantage of this point of view is that it takes into accountarbitrary dependencies between the variables. For exam-ple, if p21 = p2p3 then 2p1 d p1 = p3 d p2 + p2 d p3 . Inparticular, if the variables pi are all functions of t, as inthe previous section, then
dM =∂M
∂td t+
n∑i=1
∂M
∂pi
∂pi∂t
d t.
Dividing through by dt gives the total derivative dM / dt.
29.3 The total derivative as a linearmap
Let U ⊆ Rn be an open subset. Then a function f :U → Rm is said to be (totally) differentiable at a pointp ∈ U , if there exists a linear map d fp : Rn → Rm
(also denoted Dpf or Df(p)) such that
limx→p
∥f(x)− f(p)− d fp(x− p)∥∥x− p∥
= 0.
The linear map d fp is called the (total) derivative or(total) differential of f at p . A function is (totally)differentiable if its total derivative exists at every pointin its domain.Note that f is differentiable if and only if each of its com-ponents fi : U → R is differentiable. For this it is nec-essary, but not sufficient, that the partial derivatives ofeach function fj exist. However, if these partial deriva-tives exist and are continuous, then f is differentiable andits differential at any point is the linear map determinedby the Jacobian matrix of partial derivatives at that point.
29.4 Total differential equation
Main article: Total differential equation
A total differential equation is a differential equation ex-pressed in terms of total derivatives. Since the exteriorderivative is a natural operator, in a sense that can begiven a technical meaning, such equations are intrinsicand geometric.
98 CHAPTER 29. TOTAL DERIVATIVE
29.5 Application to equation sys-tems
In economics, it is common for the total derivative to arisein the context of a system of equations.[1]:pp. 217-220 Forexample, a simple supply-demand system might specifythe quantity q of a product demanded as a function D ofits price p and consumers’ income I, the latter being anexogenous variable, and might specify the quantity sup-plied by producers as a function S of its price and twoexogenous resource cost variables r and w. The resultingsystem of equations,
q = D(p, I),
q = S(p, r, w),
determines the market equilibrium values of the variablesp and q. The total derivative of, for example, p with re-spect to r, dp
dr , gives the sign and magnitude of the reac-tion of the market price to the exogenous variable r. Inthe indicated system, there are a total of six possible to-tal derivatives, also known in this context as comparativestatic derivatives: dp/dr, dp/dw, dp/dI, dq/dr, dq/dw, anddq/dI. The total derivatives are found by totally differ-entiating the system of equations, dividing through by,say dr, treating dq/dr and dp/dr as the unknowns, settingdI=dw=0, and solving the two totally differentiated equa-tions simultaneously, typically by using Cramer’s rule.
29.6 References[1] Chiang, Alpha C. Fundamental Methods of Mathematical
Economics, McGraw-Hill, third edition, 1984.
• A. D. Polyanin and V. F. Zaitsev, Handbook ofExact Solutions for Ordinary Differential Equations(2nd edition), Chapman & Hall/CRC Press, BocaRaton, 2003. ISBN 1-58488-297-2
• From thesaurus.maths.org total derivative
29.7 External links• Weisstein, Eric W., “Total Derivative”,MathWorld.
• http://www.sv.vt.edu/classes/ESM4714/methods/df2D.html
Chapter 30
Virtual displacement
For the particle trajectory x(t) and its virtual trajectory x′(t) ,at position x1 , time t1 , the virtual displacement is δx Thestarting and ending positions for both trajectories are at x0 andx2 respectively.
m
C
N
q
f(x,y)=0
One degree of freedom.
N
C
q 1
q 2
f(x,y,z)=0
m
q 1
q 2
Two degrees of freedom.Constraint force C and virtual displacement δr for aparticle of mass m confined to a curve. The resultantnon-constraint force is N. The components of virtualdisplacement are related by a constraint equation.
In analytical mechanics, a branch of applied mathemat-ics and physics, a virtual displacement δri “is an as-sumed infinitesimal change of system coordinates occur-ring while time is held constant. It is called virtual ratherthan real since no actual displacement can take placewithout the passage of time.”[1]:263 Also, virtual displace-ments are spatial displacements exclusively - time is fixedwhile they occur. When computing virtual differentialsof quantities that are functions of space and time coor-dinates, no dependence on time is considered (formallyequivalent to saying δt = 0).In modern terminology virtual displacement is a tangentvector to the manifold representing the constraints at afixed time. Unlike regular displacement which arisesfrom differentiating with respect to time parameter talong the path of the motion (thus pointing in the direc-tion of the motion), virtual displacement arises from dif-ferentiating with respect to the parameter ε enumeratingpaths of the motion varied in a manner consistent with theconstraints (thus pointing at a fixed time in the directiontangent to the constraining manifold). The symbol δ istraditionally used to denote the corresponding derivative
δ =∂
∂ϵ
∣∣∣∣ϵ=0
.
30.1 Comparison between virtualand actual displacements
The total differential of any set of system position vectors,ri, that are functions of other variables
q1, q2, ..., qm
and time t may be expressed as follows:[1]:264
dri =∂ri∂t
dt+m∑j=1
∂ri∂qj
dqj
99
100 CHAPTER 30. VIRTUAL DISPLACEMENT
If, instead, we want the virtual displacement (virtual dif-ferential displacement), then[1]:265
δri =m∑j=1
∂ri∂qj
δqj
This equation is used in Lagrangian mechanics to re-late generalized coordinates, qj, to virtual work, δW, andgeneralized forces, Qj.
30.2 Virtual work
In analytical mechanics the concept of a virtual displace-ment, related to the concept of virtual work, is meaning-ful only when discussing a physical system subject to con-straints on its motion. A special case of an infinitesimaldisplacement (usually notated dr), a virtual displacement(denoted δr) refers to an infinitesimal change in the po-sition coordinates of a system such that the constraintsremain satisfied.For example, if a bead is constrained to move on a hoop,its position may be represented by the position coordi-nate θ, which gives the angle at which the bead is situated.Say that the bead is at the top. Moving the bead straightupwards from its height z to a height z + dz would rep-resent one possible infinitesimal displacement, but wouldviolate the constraint. The only possible virtual displace-ment would be a displacement from the bead’s position,θ to a new position θ + δθ (where δθ could be positive ornegative).
30.3 See also• D'Alembert principle
• Virtual work
30.4 References[1] Torby, Bruce (1984). “Energy Methods”. Advanced Dy-
namics for Engineers. HRW Series in Mechanical Engi-neering. United States of America: CBS College Publish-ing. ISBN 0-03-063366-4.
30.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 101
30.5 Text and image sources, contributors, and licenses
30.5.1 Text• Generalized coordinates Source: https://en.wikipedia.org/wiki/Generalized_coordinates?oldid=678575928 Contributors: The Anome,
Rmilson, CharlesMatthews, Giftlite, BenFrantzDale, Tom harrison, Mathbot, Chobot, ChrisChiasson, YurikBot, Jimp, Larsobrien, Smack-Bot, TimBentley, WikiPedant, Valenciano, Jgates, Owlbuster, VictorAnyakin, Plasticup, VolkovBot, JohnBlackburne, Thurth, Sylviaelse,SieBot, BotMultichill, Renatops, Niceguyedc, Danmichaelo, Brews ohare, Count Truthstein, Crowsnest, Addbot, Gmcastil, ,ماني Zor-robot, Citation bot, RibotBOT, Ptjackyll, Maschen, Zueignung, Helpful Pixie Bot, Prof McCarthy, F=q(E+v^B), Hublolly, AJDaviesOUand Anonymous: 25
• Lagrangianmechanics Source: https://en.wikipedia.org/wiki/Lagrangian_mechanics?oldid=680753255 Contributors: Derek Ross, CYD,Tarquin, AstroNomer~enwiki, Andre Engels, Roadrunner, Peterlin~enwiki, Isis~enwiki, Michael Hardy, Tim Starling, Pit~enwiki, Wap-caplet, Karada, Looxix~enwiki, Stevan White, AugPi, Charles Matthews, Phys, Raul654, Robbot, Giftlite, Wolfkeeper, Lethe, Tom har-rison, Dratman, Ajgorhoe, Zhen Lin, Jason Quinn, DefLog~enwiki, Icairns, AmarChandra, CALR, R6144, Laurascudder, Chairboy,Bobo192, Haham hanuka, Helixblue, Oleg Alexandrov, Linas, StradivariusTV, Mpatel, Isnow, SeventyThree, Graham87, K3wq, Rjwilmsi,Mathbot, Chobot, ChrisChiasson, Krishnavedala, DVdm, Wavelength, RobotE, RussBot, Robert Turner, Ksyrie, SmackBot, InverseHyper-cube, KnowledgeOfSelf, KocjoBot~enwiki, Frédérick Lacasse, Silly rabbit, Colonies Chris, Jgates, Elzair, Lambiam, Xenure, Luis Sanchez,JRSpriggs, Szabolcs Nagy, Grj23, Gregbard, Cydebot, Rwmcgwier, Xaariz, Headbomb, Jomoal99, Sbandrews, JAnDbot, Hamsterlopithe-cus, Cstarknyc, Dream Focus, Pcp071098, Soulbot, JBdV, Pixel ;-), First Harmonic, ANONYMOUS COWARD0xC0DE, E104421,Hoyabird8, R'n'B, Wa3frp, ChrisfromHouston, LordAnubisBOT, Plasticup, CompuChip, Lseixas, VolkovBot, Camrn86, Philip Trueman,BertSen, Michael H 34, Aither~enwiki, StevenBell, YohanN7, Filos96, ClueBot, PipepBot, Razimantv, Laudak, Ultinate, Zen Mind, Drag-onBot, Shsteven, Mleconte, Brews ohare, Crowsnest, SilvonenBot, Addbot, Favonian, Numbo3-bot, Lightbot, OlEnglish, ,سعی Yobot,AnomieBOT, Galoubet, Materialscientist, Citation bot, Ywaz, RibotBOT, Gsard, Ct529, Kxx, Dwightfowler, Sławomir Biały, Craig Pem-berton, Tal physdancer, RedBot, Jordgette, Obsidian Soul, RjwilmsiBot, Mathematici6n, Edouard.darchimbaud, EmausBot, Manastra,Dadaist6174, JSquish, ZéroBot, Druzhnik, SporkBot, AManWithNoPlan, Maschen, ClueBot NG, Manubot, Snotbot, Helpful Pixie Bot,Jcc2011, CedricMC, Bibcode Bot, BG19bot, F=q(E+v^B), Syedosamaali786, BattyBot, Khazar2, Themeasureoftruth, Cgabe6, Hublolly,Minime12358, Mark viking, Kevincassel, Dabarton91, Sauhailam, 22merlin, Yikkayaya, Sofia Koutsouveli, Julian ceaser and Anonymous:146
• Action (physics) Source: https://en.wikipedia.org/wiki/Action_(physics)?oldid=678923324 Contributors: Bryan Derksen, The Anome,Taral, XJaM, B4hand, Michael Hardy, Tim Starling, SebastianHelm, Angela, AugPi, Charles Matthews, The Anomebot, Phys, Kwantus,Jph, Jheise, Tobias Bergemann, Filemon, Snobot, Ancheta Wis, Clementi, Giftlite, Wolfkeeper, Everyking, Dratman, Michael Devore,DefLog~enwiki, AmarChandra, Lumidek, Tomturner, Guanabot, Dbachmann, Laurascudder, Shanes, Nickj, Ntmatter, Matt McIrvin,Scentoni, Pazouzou, Dominic, Linas, Ae-a, BD2412, Tlroche, Zbxgscqf, Mathbot, Chobot, DVdm, WriterHound, Buggi22, YurikBot,KSmrq, Archelon, Asparn, E2mb0t~enwiki, BOT-Superzerocool, Divide, Digfarenough, Jaysbro, GrinBot~enwiki, Finell, SmackBot, RD-Bury, RobotJcb, Complexica, Fredvanner, Colonies Chris, RyanC., Berland, LuchoX, Mwtoews, Yevgeny Kats, John, Physis, Makyen,JRSpriggs, Gregbard, Cydebot, WillowW, Fl, Karl-H, Thijs!bot, Headbomb, EdJohnston, JAnDbot, Robert.hipple, .anacondabot, Hroðulf,Bakken, Adiel lo~enwiki, LordAnubisBOT, Idioma-bot, Cuzkatzimhut, Thurth, Venny85, Vukkarak, Antixt, Spinningspark, Cnilep, SueRangell, Kbrose, Patamia, Tugjob, MiNombreDeGuerra, ArdClose, Marsupilamov, Gulmammad, PhySusie, Alexey Muranov, GeorgeR-aetz, Ano-User, YouRang?, Pichpich, Cmfuen, Hess88, Addbot, CosmiCarl, Zorrobot, Praveen.kolluru, Luckas-bot, Yobot, Yngvadottir,AnomieBOT, LilHelpa, GrouchoBot, FrescoBot, Francis Lima, Rausch, Abc518, EmausBot, Share8smart, Chricho, Brazmyth, Ebram-bot, Quondum, JoeSperrazza, Maschen, Frietjes, Helpful Pixie Bot, Ragnarstroberg, F=q(E+v^B), Halfb1t, Johnepearson, Chink3tom,Melonkelon, Wicklet, Physikerwelt, Mario Castelán Castro, Isambard Kingdom and Anonymous: 74
• AQUAL Source: https://en.wikipedia.org/wiki/AQUAL?oldid=674753759 Contributors: Ben Standeven, BD2412, RjwilmsiBot,Maschen, Bibcode Bot and Anonymous: 2
• Averaged Lagrangian Source: https://en.wikipedia.org/wiki/Averaged_Lagrangian?oldid=674825285 Contributors: Mleconte,Crowsnest, Maschen and Bibcode Bot
• Canonical coordinates Source: https://en.wikipedia.org/wiki/Canonical_coordinates?oldid=628136130 Contributors: Jason Quinn, Lu-midek, Linas, Mathbot, YurikBot, KSmrq, Archelon, Colonies Chris, Yevgeny Kats, Steel, Ebyabe, Lseixas, Thurth, Geometry guy,Venny85, Senderista~enwiki, Count Truthstein, Szteven, Addbot, Mathieu Perrin, Xqbot, Omnipaedista, Point-set topologist, LucienBOT,Rausch, Sgoder, Maschen and Anonymous: 10
• Classical field theory Source: https://en.wikipedia.org/wiki/Classical_field_theory?oldid=674751849 Contributors: Bueller 007,Haukurth, Phys, Phil Boswell, Abdull, Rich Farmbrough, Masudr, Firsfron, Linas, Mpatel, BD2412, Rjwilmsi, YurikBot, Gaius Cor-nelius, Tong~enwiki, Dhollm, Ilmari Karonen, SmackBot, Colonies Chris, Twalton, Ligulembot, Davius, Konradek, Headbomb, Jpod2,Hodja Nasreddin, Daniele.tampieri, Lseixas, Gogobera, TXiKiBoT, Natural Philosopher, MaAnReynolds, DragonBot, Djr32, Melon-Bot, Crowsnest, Addbot, MrOllie, Favonian, Potekhin, AnomieBOT, Citation bot, Gsard, Thehelpfulbot, Reconsider the static, BrianannMacAmhlaidh, Timetraveler3.14, Maschen, Zueignung, Mhiji, Rezabot, Helpful Pixie Bot, HilbertSpace271828, Abitslow, Monkbot, Oc-towalrus, Teelaskeletor and Anonymous: 23
• Covariant classical field theory Source: https://en.wikipedia.org/wiki/Covariant_classical_field_theory?oldid=589530587 Contributors:Giftlite, Rich Farmbrough, Mpatel, Tong~enwiki, SmackBot, Twalton, Vyznev Xnebara, Dilane, Ekotkie, STBot, Red Act, Addbot, Point-set topologist, Gsard, Rausch, Clessig, Maschen, Helpful Pixie Bot and Anonymous: 3
• D'Alembert’s principle Source: https://en.wikipedia.org/wiki/D'Alembert’s_principle?oldid=678918288 Contributors: The Anome,XJaM, Heron, Patrick, Michael Hardy, Ojigiri~enwiki, Tobias Bergemann, Ancheta Wis, Decumanus, BenFrantzDale, Jason Quinn, Æ,Eyrian, Oleg Alexandrov, Linas, Btyner, Milez, Mathbot, ChrisChiasson, Krishnavedala, Sanpaz, Roboto de Ajvol, YurikBot, RussBot,Buster79, Larsobrien, Teply, SmackBot, Harald88, Gilliam, Silly rabbit, E4mmacro, Nakon, Mets501, Geltron, Friendlystar, Gregbard,Nilfanion, Headbomb, BokicaK, JAnDbot, EagleFan, R'n'B, Elchafa, Plasticup, Rudolf33, DorganBot, Thurth, TXiKiBoT, Jameyshaef,Kormi~enwiki, KoenDelaere, OKBot, Mild Bill Hiccup, Brews ohare, Crowsnest, Addbot, LaaknorBot, Spra, Aviados, Ettrig, StarLight,Luckas-bot, Yobot, Ptbotgourou, THEN WHO WAS PHONE?, Citation bot, ArthurBot, Martin Kraus, Archaeodontosaurus, Vrenator,EmausBot, RRosenkjar, ClueBot NG, Eml5526.s11.team4.cook, Mattte, Prof McCarthy, Solomon7968, CitationCleanerBot, MrinmoySengupta, Vanamonde93, AKSoni13 and Anonymous: 46
102 CHAPTER 30. VIRTUAL DISPLACEMENT
• Fiber derivative Source: https://en.wikipedia.org/wiki/Fiber_derivative?oldid=601738783 Contributors: Bearcat, Xezbeth, David Epp-stein, Trevorgoodchild, Yobot, Jesse V. and Qetuth
• FLEXPART Source: https://en.wikipedia.org/wiki/FLEXPART?oldid=662920758 Contributors: Postcard Cathy, BG19bot, Iaritmioawpand Gup00
• Generalized forces Source: https://en.wikipedia.org/wiki/Generalized_forces?oldid=567381556Contributors: ChrisChiasson, SmackBot,Radagast83, Alaibot, Qrystal, JCarlos, VolkovBot, TXiKiBoT, Addbot, Yobot, RedBot, Finemann, Prof McCarthy and Anonymous: 2
• Geometric mechanics Source: https://en.wikipedia.org/wiki/Geometric_mechanics?oldid=662708406 Contributors: Michael Hardy,Bearcat, Rjwilmsi, David Eppstein, Yobot, Josve05a, Maschen and Jamontaldi
• Gibbons–Hawking–York boundary term Source: https://en.wikipedia.org/wiki/Gibbons%E2%80%93Hawking%E2%80%93York_boundary_term?oldid=618834079 Contributors: Bryan Derksen, Michael Hardy, Jason Quinn, Drbogdan, Rjwilmsi, Lionelbrits, YevgenyKats, Headbomb, Lantonov, Shawn in Montreal, DOI bot, Legobot, Yobot, Omnipaedista, M5brane, Raidr, GoldPhoenix99, Mathew-Townsend and Anonymous: 4
• Hamilton’s principle Source: https://en.wikipedia.org/wiki/Hamilton’s_principle?oldid=679939147 Contributors: Michael Hardy,Tango, SebastianHelm, Robbot, Steuard, BozMo, Army1987, Linas, Btyner, Michielsen, SmackBot, Jan Bielawski, Kripkenstein, JR-Spriggs, Eli84, WillowW, Thijs!bot, Headbomb, ThomasPusch, JAnDbot, First Harmonic, Ceccorossi~enwiki, Thurth, Crowsnest, Ad-dbot, Fgnievinski, Ettrig, Luckas-bot, Yobot, Amirobot, AnomieBOT, ^musaz, Materialscientist, Comt Till, Ysyoon, Chricho, 1cycyc1,Maschen, Zfeinst, F=q(E+v^B), ITolib, CarrieVS, Couplingconstant, SteenthIWbot, Mark viking, Faizan, Kevincassel and Anonymous: 16
• Inverse problem for Lagrangianmechanics Source: https://en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics?oldid=674740275 Contributors: Michael Hardy, Charles Matthews, Oleg Alexandrov, Hmains, TheTito, Sullivan.t.j, DavidCBryant, Pjoef, DOIbot, Yobot, Citation bot 1, Maschen, CitationCleanerBot, CarrieVS, DG-on-WP and Anonymous: 1
• Jacobi coordinates Source: https://en.wikipedia.org/wiki/Jacobi_coordinates?oldid=670662006 Contributors: Michael Hardy, CBM,MichaelSchoenitzer, Brews ohare, Addbot, Luckas-bot, Erik9bot, Tom.Reding, Teapeat, Helpful Pixie Bot, Justincheng12345-bot, Rfass-bind and Anonymous: 1
• Joseph-Louis Lagrange Source: https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange?oldid=678463431 Contributors: Derek Ross,Bryan Derksen, Zundark, The Anome, Gareth Owen, Danny, XJaM, Camembert, Olivier, Michael Hardy, Nixdorf, Gabbe, Komap,Alfio, Stw, Looxix~enwiki, ArnoLagrange, Ellywa, Stevan White, Docu, Theresa knott, Snoyes, Angela, Pizza Puzzle, Jengod, Re-volver, Charles Matthews, Reddi, Maximus Rex, MiLo28, Kwantus, Huangdi, Robbot, Jaredwf, Halibutt, Snobot, Giftlite, Wwoods,Everyking, Curps, JeffBobFrank, Kyl191, Wmahan, Utcursch, Ynh~enwiki, Ary29, GeoGreg, Picapica, D6, Discospinster, Eb.hoop,Rich Farmbrough, Dbachmann, Paul August, Bender235, Ignignot, Laurascudder, Bill Thayer, Bobo192, Army1987, Wood Thrush,I9Q79oL78KiL0QTFHgyc, Obradovic Goran, Wrs1864, MPerel, Haham hanuka, LutzL, Espoo, Jumbuck, Alansohn, Eric Kvaalen,Andrewpmk, AzaToth, Comrade009, Bart133, VivaEmilyDavies, Derbeth, Endersdouble, Wakowill, BerserkerBen, Oleg Alexandrov,Yousaf465, Woohookitty, Carcharoth, Emerson7, Mandarax, Magister Mathematicae, Rjwilmsi, HannsEwald, Olessi, Kasparov, Rangek,Greg321, Crazycomputers, Rbonvall, Numa, Carrionluggage, Russavia, Chobot, Gdrbot, YurikBot, Wavelength, RattusMaximus, Nei-therday, RussBot, Sillybilly, Grafen, UVW, Dethomas, Tanet, Zzuuzz, Alasdair, HereToHelp, Curpsbot-unicodify, Apapa, DVD R W,
robot, Attilios, SmackBot, Smitz, Hydrogen Iodide, Ze miguel, KocjoBot~enwiki, Eskimbot, Jab843, Commander Keane bot, Tim-bouctou, Dahn, Jjalexand, Alan smithee, MalafayaBot, Silly rabbit, Dlohcierekim’s sock, Nbarth, DHN-bot~enwiki, Darth Panda, CharlesMoss, VanishedUser 0001, Lesnail, Akriasas, Jbergquist, TenPoundHammer, Drengor, Nishkid64, Hemmingsen, BillFlis, Michael Greiner,Inquisitus, Math man, Alessandro57, Shoeofdeath, CRGreathouse, BoH, Drinibot, Charvex, Myasuda, Asztal, Yaris678, Doctormatt, Cy-debot, Grahamec, Pascal.Tesson, Studerby, Englishnerd, Plch, Thijs!bot, Epbr123, Imusade, Bcent1234, AntiVandalBot, RobotG, RaduBorza, Activist, Hannes Eder, Spencer, Storkk, Myanw, Bèrto 'd Sèra, Gökhan, JAnDbot, Deflective, Barek, Db099221, PhilKnight, DreamFocus, .anacondabot, Magioladitis, VoABot II, Catslash, David Eppstein, Kraxler, Pax:Vobiscum, Nzv8fan, Nono64, J.delanoy, Skeptic2,SuperGirl, Hans Dunkelberg, KeepItClean, McSly, NewEnglandYankee, Rpeh, VolkovBot, ABF, URBIS, Satani, Nousernamesleft, A4bot,Jobu0101, Schafesd, Ferengi, BigZam85, MajorHazard, Spitfire8520, Arcfrk, Logan, Resurgent insurgent, Haiviet~enwiki, GirasoleDE,SieBot, Mikemoral, YonaBot, Caltas, Elakhna, Monegasque, Chansonh, AlanUS, Vojvodaen, Vandalfighter101, Elassint, ClueBot, TheThing That Should Not Be, All Hallow’sWraith, Unbuttered Parsnip, Tomas e, Puchiko, DragonBot, Excirial, Versus22, Crowsnest, DumZ-iBoT, Palnot, RogDel, Gerhardvalentin, NellieBly, MystBot, Addbot, Jkasd, Fyrael, CarsracBot, SamatBot, Tide rolls, Lightbot, Zorrobot,Վազգեն, DrFO.Tn.Bot, Legobot, Luckas-bot, Yobot, Amble, KamikazeBot, Tempodivalse, AnomieBOT, Rubinbot, Götz, Sz-iwbot,ArthurBot, Nifky?, Xqbot, Random astronomer, Gwelch, Oxy86, FerranMir, Almabot, Omnipaedista, RibotBOT, Saalstin, Amaury, Char-vest, , Green Cardamom, Magnagr, LucienBOT, Pepper, Chenopodiaceous, Tkuvho, Plucas58, Michaeltheish, Tom.Reding, Skyerise,RedBot, Grungey baby, CLC Editorial, DixonDBot, Joshua L. O'Brien, Epfr, Allen65, DARTH SIDIOUS 2, RjwilmsiBot, Regancy42,EmausBot, Stryn, Wikipelli, Auró, JSquish, Kkm010, 15turnsm, ZéroBot, Fæ, Traxs7, D.Lazard, Emilyhenslerinventedmath, Arman Ca-gle, JeanneMish, Carmichael, Sapphorain, ChuispastonBot, DASHBotAV, ResearchRave, ClueBot NG, Mgvongoeden, HazelAB, Widr,Asalrifai, Helpful Pixie Bot, SzMithrandir, Ramaksoud2000, MusikAnimal, Snow Blizzard, Brad7777, Renwu66, Chip123456, JeanM,BattyBot, Doyoon1995, Ninmacer20, Heterodoxa, Radisapfc, LuoSciOly, Webclient101, Symplectic.Nova, Aries no Mur, Lugia2453, VI-AFbot, Nimetapoeg, Mark viking, Purnendu Karmakar, Tentinator, EFZR090440, Jianhui67, JAaron95, OccultZone, Brettdsteele, Cnejatoc, Monkbot, Teddyktchan, Danstronger, KasparBot and Anonymous: 297
• Lagrangian point Source: https://en.wikipedia.org/wiki/Lagrangian_point?oldid=678482287 Contributors: Damian Yerrick, MagnusManske, Paul Drye, The Epopt, Ansible, Brion VIBBER, Mav, Bryan Derksen, Zundark, Tarquin, Wayne Hardman, XJaM, JeLuF,PierreAbbat, Roadrunner, Camembert, Bdesham, Patrick, JohnOwens, Michael Hardy, Frank Shearar, Mcarling, Skysmith, (, Sebas-tianHelm, Minesweeper, Ahoerstemeier, Muriel Gottrop~enwiki, Caid Raspa, Julesd, Whkoh, Cherkash, Emperorbma, Jitse Niesen,Audin, Doradus, Wik, Paul-L~enwiki, Nickshanks, Pakaran, Finlay McWalter, Robbot, Astronautics~enwiki, Pigsonthewing, Sverdrup,Rorro, Kielsky, Rursus, Bkell, Netjeff, HaeB, GreatWhiteNortherner, Jason in MN, Giftlite, JamesMLane, DocWatson42, DavidCary,Wolfkeeper, Art Carlson, MSGJ, Herbee, Angmering, Wwoods, Curps, Pascal666, Golbez, Cam, Zeimusu, ConradPino, Fpahl, Jo-damiller, Beland, Mako098765, OrangUtanUK, WhiteDragon, Mzajac, OwenBlacker, CanSpice, Tomwalden, Rich Farmbrough, Gua-nabot, Sladen, Vsmith, Florian Blaschke, ArnoldReinhold, Bo Lindbergh, Bender235, Kbh3rd, Karmafist, Kwamikagami, Kross, Tver-beek, PhilHibbs, Shanes, Dennis Brown, Euyyn, Mike Schwartz, Matt McIrvin, Larry V, Jumbuck, Grutness, Gary, Eric Kvaalen,Arthena, Apoc2400, RoySmith, Mlm42, Hu, Gomi no sensei, Dschwen, Suruena, Evil Monkey, Staeiou, Kindalas, Cmprince, Pauli133,Axeman89, LukeSurl, Kitch, BerserkerBen, ReelExterminator, Agingjb, WilliamKF, Woohookitty, Natcase, Duncan.france, CharlesC,Fxer, Zooks527, Mandarax, Graham87, Protargol, Rjwilmsi, Zbxgscqf, Vary, Strait, Bubba73, Nguyen Thanh Quang, Dracontes, Cjpuf-fin, Mathbot, Gparker, Quuxplusone, Chobot, PointedEars, Neitherday, Hairy Dude, Jimp, 4C~enwiki, RussBot, Chuck Carroll, Hell-
30.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 103
bus, Hydrargyrum, Ergzay, Johnny Pez, The Merciful, Robertvan1, Aeusoes1, RazorICE, BlackAndy, Raven4x4x, Crasshopper, Cycle-skinla, Deeday-UK, Wikicheng, Chase me ladies, I'm the Cavalry, Closedmouth, Reyk, Back ache, Geoffrey.landis, HereToHelp, Skit-tle, Diegorodriguez~enwiki, Cmglee, Ffangs, Hide&Reason, SmackBot, Unschool, Robotbeat, Incnis Mrsi, Herostratus, C.Fred, Declare,Brianski, BirdValiant, Grokmoo, GovanRear, Salvo46, Riaanvn, Dreg743, Jprg1966, Thumperward, Rpspeck, Scienz Guy, Hgrosser,Audriusa, WDGraham, Trekphiler, Tamfang, Writtenright, MattHanlon, Spacecaldwell, Allan McInnes, Subheight640, TheLimbicOne,Romanski, Vina-iwbot~enwiki, Lambiam, JorisvS, Terry Bollinger, 5telios, Rsquid, Loadmaster, TheHYPO, Dicklyon, Megane~enwiki,EdC~enwiki, MrDolomite, TJ Spyke, Craigboy, Tmangray, Benabik, Hamish.MacEwan, Memetics, Ruslik0, N2e, Goatchurch, NecessaryEvil, Cydebot, Peripitus, Savre, Reywas92, VAXHeadroom, LarryMColeman, Clh288, Christian75, Thijs!bot, Skiltao, AndrewDressel,Undomiel, Uruiamme, Noclevername, 1-54-24, Hawk16zz, Saimhe, Vendettax, Yellowdesk, Tulio22, Ingolfson, JAnDbot, Deflective,CosineKitty, Medconn, Chromosome, ReK, Mwarren us, 100110100, Dream Focus, Rollred15, Seanette, WolfmanSF, QuizzicalBee,Mrld, SHCarter, LosD, Swpb, RypER, Thermal0xidizer, Originalname37, JMyrleFuller, Just James, JaGa, Onigame, Alfachino, Gwern,Atarr, Jim.henderson, Sm8900, CommonsDelinker, Kevin Carmody, AstroHurricane001, Hans Dunkelberg, ChrisfromHouston, Sidhekin,Tdadamemd, Andrew Moise, Dargaud, Rod57, JavaJawaUK, Tarotcards, DjScrawl, Ohms law, Atropos235, Black Walnut, Pborenstein,Speciate, Xenonice, VolkovBot, Larryisgood, Pleasantville, Mrh30, Otherdumbname, Seattle Skier, Grammarmonger, Larry R. Holmgren,Sdsds, KimiSan, Phoenix2602, Baileypalblue, OlavN, BeIsKr, Dependent Variable, AmouDaria, Taho s, Friscorose, Peregrinoerick, Alle-borgoBot, Kktor, BillHicksRulez, Hughey, 1Shaggy1, Coronellian~enwiki, SieBot, WereSpielChequers, Alex Essilfie, Scimike, KGyST,Agesworth, K.h.w.m, Paolo.dL, Roger.ritenour, Beast of traal, Sailing rich, Roujo, RSStockdale, Anchor Link Bot, PlantTrees, Mar-tarius, ArdClose, Number774, CounterVandalismBot, Piledhigheranddeeper, Deselliers, Lfast321, Brews ohare, Kittylyst, Brilliant Peb-ble, JKeck, Mchaddock, Ophello, Nathan Johnson, Astrofreak92, Feyrauth, Subversive.sound, Addbot, Immortal Imp, Roentgenium111,Llesslie, Obsidianspider, AnnaFrance, 84user, Numbo3-bot, Bigzteve, Lightbot, Legobot, Luckas-bot, Yobot, MarioS, Amirobot, Alde-baran66, AnomieBOT, JackieBot, Flewis, Citation bot, DirlBot, LilHelpa, Xqbot, DrRevXyzzy, Deeptext, Invent2HelpAll, Stevenenglund,Metafax1, Ataleh, Lidmann, 11cookeaw1, GliderMaven, FrescoBot, Wtthompson, Liiiii, EmilTyf, Kvidre!, Neilksomething, AstaBOTh15,Dcshank, DanteEspinoza1989, Tom.Reding, LagrangeCalvert, TeigeRyan, Cnwilliams, Meier99, Topaxi, TobeBot, Dinamik-bot, Rjwilm-siBot, TjBot, Mathematici6n, Androstachys, Tesseract2, EmausBot, Lucien504, Mmeijeri, FlyAkwa, Hhhippo, Listmeister, ZéroBot, Ri-ommar, Crua9, Mattedia, Δ, Giulianone, SkywalkerPL, RockMagnetist, Dkoz314, Teapeat, Karthikrrd, Fjörgynn, ClueBot NG, Giggett,Matsaball, Macarenses, Gd199210018, Muon, MobyDick71, Bibcode Bot, Gauravjuvekar, Doyna Yar, BG19bot, Rasheeq1, Anwalt-frankfurt, Wc620, Trevayne08, Zedshort, Gianamar, Cliff12345, Ibk4173hf3p, BattyBot, Fraulein451, Dexbot, Olly314, Hms1103, Lu-gia2453, Tony Mach, Andyhowlett, Tommy.Hudec, Reatlas, Rfassbind, Tbrien88, Wikiuser314159, RedZay7, Itaspence, Samatict, Alex-eiVronsky, DJHerls, Monkbot, Warqer, 0Archimedes0, Pabnau, TuxedoMonkey, Tetra quark, Appable, Kahntagious, KasparBot, MartinColes, Eric1024 and Anonymous: 350
• Lagrangian system Source: https://en.wikipedia.org/wiki/Lagrangian_system?oldid=680382102 Contributors: Michael Hardy, Man-darax, InverseHypercube, Widefox, YohanN7, Yobot, AnomieBOT, Gsard, Dewritech, Maschen, Helpful Pixie Bot, KLBot2, BG19botand Anonymous: 1
• Minimal coupling Source: https://en.wikipedia.org/wiki/Minimal_coupling?oldid=677161402 Contributors: Concertmusic, Schreiber-Bike, Addbot, Tassedethe, Arbitrarily0, Yobot, FrescoBot, RedBot, Maschen, Anagogist, Helpful Pixie Bot, Djmaity, Dzustin, Saehry andAnonymous: 3
• Monogenic system Source: https://en.wikipedia.org/wiki/Monogenic_system?oldid=644042412 Contributors: Michael Hardy, Or-angemike, Jason Quinn, Art LaPella, Jpbowen, Ms2ger, SmackBot, MaxEnt, R'n'B, Thurth, Addbot, StarLight, Yobot, RjwilmsiBot,Quondum, Helpful Pixie Bot and Anonymous: 1
• Ostrogradsky instability Source: https://en.wikipedia.org/wiki/Ostrogradsky_instability?oldid=674706078 Contributors: John of Read-ing, Maschen, Anagogist, Zhantongz and Anonymous: 1
• Palatini variation Source: https://en.wikipedia.org/wiki/Palatini_variation?oldid=674748776 Contributors: Charles Matthews, Ben-der235, Encyclops, IceCreamAntisocial, Alaibot, Bdr9, Jessicapierce, Bufrost, General Epitaph, Maschen, Mgvongoeden and Anonymous:4
• Rayleigh dissipation function Source: https://en.wikipedia.org/wiki/Rayleigh_dissipation_function?oldid=588359826 Contributors: Ja-son Quinn, Alvin Seville and Antiqueight
• Rheonomous Source: https://en.wikipedia.org/wiki/Rheonomous?oldid=579048699 Contributors: Samw, Jason Quinn, Camw, Cm-drObot, Oreo Priest, Sagi Harel, R'n'B, VolkovBot, Thurth, AlleborgoBot, Addbot, Yobot, Citation bot, MastiBot, Ripchip Bot, Hhhippo,Helpful Pixie Bot, KLBot2, Hamoudafg and Anonymous: 2
• Scleronomous Source: https://en.wikipedia.org/wiki/Scleronomous?oldid=655499006 Contributors: Samw, CmdrObot, Oreo Priest, SagiHarel, Thurth, Red Act, Dthomsen8, Addbot, Yobot, Citation bot, Omnipaedista, , FrescoBot, Tal physdancer, Michael.M.Chan, HelpfulPixie Bot, Hamoudafg and Anonymous: 5
• Tautological one-form Source: https://en.wikipedia.org/wiki/Tautological_one-form?oldid=601135889 Contributors: Robbot, Giftlite,Fropuff, CALR, CanisRufus, Linas, YurikBot, Orthografer, Nbarth, Ebyabe, Konradek, YKTimes, Sullivan.t.j, Stephane Redon, Haseldon,XCelam, Red Act, Geometry guy, Addbot, Mathieu Perrin, Ht686rg90, Gioprof and Anonymous: 10
• Total derivative Source: https://en.wikipedia.org/wiki/Total_derivative?oldid=674733905 Contributors: The Anome, Michael Hardy,TakuyaMurata, Silverfish, Charles Matthews, Jitse Niesen, MathMartin, Giftlite, BenFrantzDale, Dratman, Vsb, TedPavlic, MisterSheik,Peter M Gerdes, Vuo, Jonathan48, Grammarbot, MarkHudson, MarSch, Andrei Polyanin, ChrisChiasson, YurikBot, Borgx, Dpakoha,Caiyu, Lucasreddinger, Darrel francis, Erik Sandberg, Maksim-e~enwiki, Kelvie, Colonies Chris, Jim.belk, CmdrObot, Cydebot, WISo,Πrate, JAnDbot, Txomin, Magioladitis, Baccyak4H, Ac44ck, Quocminh9, Hanacy, Trigamma, Geometry guy, Addbot, Jarble, Luckas-bot,AnomieBOT, ^musaz, Rb88guy, Raulshc, Plasticspork, Duoduoduo, Mmm333k, TonyMath, Maschen, ClueBot NG, CocuBot, Frietjes,Kasirbot, Mn-imhotep, Falktan, LaurensDorival, Loraof and Anonymous: 50
• Virtual displacement Source: https://en.wikipedia.org/wiki/Virtual_displacement?oldid=679179294 Contributors: Asparagus, Ben-FrantzDale, Dratman, Oleg Alexandrov, Milez, Volfy, ChrisChiasson, Grafen, Cleared as filed, SmackBot, Bluebot, Jim.belk, JanBielawski,CmdrObot, Alaibot, Escarbot, Thurth, TXiKiBoT, ClueBot, Nick84, Addbot, Mpfiz, Yobot, , RedBot, ZéroBot, Maschen, KLBot2,ChrisGualtieri, Hamoudafg and Anonymous: 8
104 CHAPTER 30. VIRTUAL DISPLACEMENT
30.5.2 Images• File:3_particles_in_a_constrained_surface.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/3_particles_in_a_
constrained_surface.svg License: CC0 Contributors: Own work Original artist: Maschen• File:ACE_at_L1.png Source: https://upload.wikimedia.org/wikipedia/commons/7/71/ACE_at_L1.png License: Public domain Contrib-utors: http://helios.gsfc.nasa.gov/ace/gallery.html Original artist: Unknown
• File:Alembert.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Alembert.jpg License: Public domain Contributors:Didier Descouens 2002 Original artist: Maurice Quentin de La Tour
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• File:Bead_on_wire_constraint.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8e/Bead_on_wire_constraint.svg Li-cense: CC0 Contributors: Own work Original artist: Maschen
• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Originalartist: ?
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• File:Constraint_force_virtual_displacement_2_dof.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/85/Constraint_force_virtual_displacement_2_dof.svg License: CC0 Contributors: Own work Original artist: Maschen
• File:Dalembert_example.JPG Source: https://upload.wikimedia.org/wikipedia/commons/8/80/Dalembert_example.JPG License: CCBY-SA 4.0 Contributors: Own work Original artist: AKSoni13
• File:Double-Pendulum.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/78/Double-Pendulum.svg License: CC-BY-SA-3.0 Contributors: ? Original artist: User JabberWok on en.wikipedia
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• File:GodfreyKneller-IsaacNewton-1689.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg License: Public domain Contributors: http://www.newton.cam.ac.uk/art/portrait.html Orig-inal artist: Sir Godfrey Kneller
• File:Jacobi_coordinates_—_illustration_for_four_bodies.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/Jacobi_coordinates_%E2%80%94_illustration_for_four_bodies.svg License: Public domain Contributors: This file was derived from:Four-body Jacobi coordinates.JPGOriginal artist:
• Brews ohare• File:L4_diagram.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/78/L4_diagram.svg License: CC-BY-SA-3.0 Contrib-utors: ? Original artist: ?
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• File:Lagrange_points2.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ee/Lagrange_points2.svg License: CC BY 3.0Contributors:
• Lagrange_points.jpg Original artist: Lagrange_points.jpg: created by• File:Lagrange_points_simple.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/Lagrange_points_simple.svg License:
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main Contributors: ? Original artist: ?• File:Least_action_principle.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Least_action_principle.svg License:
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• File:Non_generalized_coordinates_closed_curved_path_2d_1df.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c2/Non_generalized_coordinates_closed_curved_path_2d_1df.svg License: CC0 Contributors: Own work Original artist: Maschen
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• File:PendulumWithMovableSupport.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a7/PendulumWithMovableSupport.svg License: CC-BY-SA-3.0 Contributors: Own work Original artist: New version by RubberDuck ( • )
• File:Pendulum_constraint.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/62/Pendulum_constraint.svg License: CC0Contributors: Own work Original artist: Maschen
• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007
• File:Rigid_Body_in_Free_Fall_with_Angular_Velocity.JPG Source: https://upload.wikimedia.org/wikipedia/commons/2/25/Rigid_Body_in_Free_Fall_with_Angular_Velocity.JPG License: CC BY-SA 4.0 Contributors: Own work Original artist: AKSoni13
• File:RocketSunIcon.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d6/RocketSunIcon.svg License: Copyrighted freeuse Contributors: Self made, based on File:Spaceship and the Sun.jpg Original artist: Me
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• File:SimplePendulum01.JPG Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/SimplePendulum01.JPG License: CC-BY-SA-3.0 Contributors: User:Helix84; Original artist: The original uploader was realname at Chinese Wikipedia
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