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action function 42adapted chart to a distribution 24alternative Lagrangians 123Anderson-Duchamp-Krupka conditions 6 Iangular momentum 159- conservation law 159, 165anholonomic prolongation of a fibered
manifold 30annihilator 2 IAristotle group 159associated- fiber chart 30- fibered coordinates 30, 3I- system of a closed two-form 25autonomous-Lagrangean system 71,210- serni-finslerian structure 223base of a fibered manifold 29brachystochrone 3, 4-problem 3bunch of final constraint submanifolds 133bundle of JT,-vertical vectors 31canonical- connection 222- dynamical two-form 221canonical form- of Hamilton equations 81, 107, 108, 145- of a Lagrangean system 67- of a Lepagean 2-form 67canonical transformation 187Cartan- connection 224- distribution 34, 37, 39- form 8, 47, 48, 209,- Theorem 26- symmetry 172Cauchy characteristic 25, 191- vector field 25, 191characteristic distribution- of a closed two-form 25, 175- of a Lagrangean system 88, 107, 108, 146Christoffel symbols 121,224
Index
classical- Hamilton-Jacobi equation 18, 174, 189- Hamilton-Jacobi integration method 18,
197classification- of Lagrangean systems 130- of symmetries 150- of variational semispray connections 227codistribution 22compatibility conditions 38complementary distributions 21complete integral 18, 174, 175- of a closed 2-form 176complete set of independent first integrals23,26
completely integrable- connection 38- distribution 23, 207- generalized connection 40components of ajet field 38configuration space 31, 63, 15Iconnection on a fibered manifold 37, 38,
183,195conservation law 149, 152, 157constant of the motion 149, 152, 160,constrained system J30constraint algorithm 25, 130, 131, 132,
134,172,210contact- form 34- transformation 212contactization 34contactness 32continuous distribution 21Coordinate free Liouville Theorem 179Coriolis force 14cosymplectic Lagrangean system 99, 210cotangent representation of symplectic
Lagrangean systems 218critical section of a Lagrangian 50, 51curvature of a connection 38Darboux chart 26, 175
Darboux Theorem 26, 175, 176Decomposition Theorem 35deformation of a section 42, 49degenerate Lagrangean system 129degree of freedom 12, 78Dirac theory of constrained systems 103,
113,134,218,219distribution 21- of a constant rank 21, 22- of a locally constant rank 22- of a non-constrant rank 22- of class C' 21dynamical- form 37, 44, 53, 116- symmetry 151, 171eikonal equation 191Embedding Theorem 191energy- conservation law 159, 165- one-form 211equivalent- dynamical forms 117, 126- Lagrangians 10, 61, 149- systems of a Hamiltonian and momenta
70Euclidean group 159,162,162,165,168,
169Euler-Lagrange distribution 90, 107, 108,
146,147- constrained to a submanifold 133Euler-Lagrange- equations I, 2, 2, 12, 41, 49, 51, 53, 80,
209- expressions 3, 46- form 41, 46, 49, 209Euler-Lagrange mapping 41, 46, 64even-order- Lagrangean system 100, 105- variational equations 105, 183, 195evolution of a function 100, 107, 109,
214exact integration methods for variational
equations 148, 149, 174extended-dynamics81,87,129,132,148,217- Lagrangian 85, 88, 108, 109extremal- of a Lagrangian 50, 51, 53,- of a locally variational form 53fiber chart 29
Index 247
fibered manifold 29field- of Lagrangean submanifolds 181- of extremals 174, 190, 192final constraint submanifold 133, 218finite symmetry 24Finsler manifold 220, 220, 223finslerian structure 223first integral 18,23, 100, 149, 152- of a Lagrangean system 152first integrals in involution 175, 177, 178first-order Lagrangean system 74, 75, 101,
102,118,122, 145,first variation 2, 41- formula 2, 41, 42, 43, 47- of the action function 42First Theorem of Emmy Noether 157flat- connection 38- generalized connection 40foliation 23force on a fibered manifold 225free- energy 12- particle on a semi-finslerian manifold 225Frobenius Theorem 24Frolicher-Nijenhuis bracket 38fundamental Lepagean equivalent 49Galilei-group 159, 162, 166,167,168,169- transformation 159, 163gauge function 61, 68General Liouville-Jacobi Theorem 189generalized-connection 37,40, 141,181,181,183- coordinates 12, 78- dynamical symmetry 172- Hamilton equations 84, 86, 88, 91, 130,
145,191,210- Hamilton-Jacobi equation 180, 189- horizontal form 40- jet field 40- Legendre chart 145- Legendre coordinates 145, 214- Legendre transformation 130, 145, 148,
214- Liouville Theorem 178, 179- Noether Theorem 160, 160, 172- Poincre-Cartan form 49- Van Hove Theorem 191
248
generating function- of a canonical transformation 174, 187
188- of a contact transformation 212generators of a distribution 21geodesics 10geodesic distance- in a field of extremals 200- in a Hamilton-Jacobi field 200geometric classification of Lagrangean
systems 96, 129, 130global- extended dynamics 131- inverse problem of the calculus of
variations 64, 65- Lagrangian 54, 65- proper dynamics 131globally variational form 53, 54Grifone connection 223Hamilton equations 17, 80, 84, 107, 108,
145,210,211,213- for autonomous Lagrangean systems 211- for semiregular autonomous Lagrangean
systems 212Hamilton- extremal 84, 129, 194, 212- form 84, 107, 108- function 17,67,70,71,210,210- vector field 90, 191, 214Hamilton's variational principle 12, 13, 15Hamilton-Jacobi- distribution 192, 193, 194, 195, 196- equation 18, 187- equation for fields of extremals 190- field 193, 194, 195- integration method 18, 174, 182, 197Hamiltonian 17,67,70,71,210,210- formalism 82, 217- system 84, 88, 100, 106Helmholtz conditions 5, 8, 57, 61, 64, 118,
161higher-order semispray problem 96, 129,
132,134,141,212Hilbert-Einstein Lagrangian 113holonomic- constraints 12, 77- Hamilton extremal 86, 87, 93, 98- section 30, 34homomorphism of fibered manifolds 30horizontal distribution 32, 37, 38
Index
horizontal form 32- of a connection 38horizontality 32horizontalization 33image of the Euler-Lagrange mapping 64independent first integrals 23, 175infinitesimal- first variation formula 47- symmetry 24integral first variation formula 47integral manifold 23- of maximal dimension 23integral mapping of a distribution 22- of maximal dimension 23integral section- of a connection 38- of a dynamical form 53, 116- of an ideal 189- of a jet field 37integrating factor I 18, 126integration- by quadratures 18, 175-methods 16, 18,142,148160,174,188,
195,207- problem 18,95,129,132,149,157,174invariant transformation- of a differential form 26-- of a distribution 24inverse Legendre transformation 106inverse problem of the calculus of
variations 4, 9, 16, 61, 65, I 18, 123,126, 150, 180, 207
- for distributions 123isomorphism of a fibered manifold 30, 75isoperimetric problem IJacobi chart 183, 196Jacobi complete integral 183, 196Jacobi Theorem 180, 182, 183- for even-order variational equations 183- for odd-order variational equations 184- for regular even-order variational
equations 186jet field 37jet manifold 29jet prolongation of a fibered manifold 29k-contact form 35kernel of the Euler-Lagrange mapping 64kinetic energy 12, 122, 126, 163, 165- of a metric 221Lagrange function I, 41, 210
Lagrange space 220, 223Lagrangean foliation 181Lagrangean submanifold 181, 196Lagrangean system 63, 81, 83, 97,126,149- admitting generalized Legendre
transformations 144,214,218- of order zero 73,75,83, 88,90, 100, 101,
102,109,127- with dynamical constraints 2 I2-- with no dynamical constraints 153- with no primary dynamical constraints
131- with no primary semispray constraints
132- with no secondary semispray constraints
133- with primary dynamical constraints 131- with primary semispray constraints 131- with secondary semispray constraints 133Lagrangian 37,41,210- affine in accelerations 74, 102- affine in velocities 73, 100, 101, 109- on R x T'M 210Legendre- chart 105-coordinates 17, 105, 148,213-transformation 17,97, 103, 105, 144,
148,213Lepage mapping- of the first kind 46- of the second kind 58Lepagean equivalent- of a locally variational form 58- of a Lagrangian 46Lepagean- n-form 41,48- one-form 41, 46, 56- two-form 55, 55, 55, 8I, 83, 97, 150,
171,208,209Levi-Civita connection 224line element induced by a minimal-order
Lagrangian 200Liouville- form 21 I- integration method 18, 207- Theorem 175, 178, 179, 182local inverse problem of the calculus of
variations 6 I, 98, 118local Lagrangian 54, 55- section of a fibered manifold 29
Index 249
locally variational form 53, 54, 8 I, I 18,161,220
Lorentz force 14, 122, 226me-parameter family of Hamilton-Jacobi
distributions 197manifold- of s-jets of local sections 29- of s-velocities 3Imaximal- integral manifold 23- vertical distribution 32mechanical system 117, 126, 225- on a semi-finslerian manifold 225method of canonical transformations 207metric on R x T M 220minimal-order Lagrangian 68, 158Minkowski- metric 208- space-time 170momentum 17,67,70,144- conservation law 159, 165- map 150multiplier I 18Newton equations 54, 80, 12INewtonian determinism 99Noether- equation 155, 161- symmetry 172- Theorem 157, 158, 172Noether-Bessel-Hagen equation 155, 16INoetherian symmetry 172non-holonomic constraints 78nonlinear connection 37odd-order- differential equations 53- Euler-Lagrange equations 68, 100, 102,
103,105,184- Lagrangean system 68, 103, 105, 183,
186,195,197,198,198,217- variational equations 68, 100, 102, 103,
105, 184one-contact form 35one-parameter family of sections 49order- of a Lagrangean system 63, 63- of contactness 36order-reduction problem for Lagrangians56,64,64
Ostrogradskii Hamiltonian 71, 85, 88Ostrogradskii momentum 71 , 85, 88
250
path of a connection 38phase space 63, 189Poincare- group 170-Lemma 36Poincare-Canan form 47point symmetry 151Poisson- bracket 27,213- Theorem 27potential energy 12, 12, 122, 165, 227potential force 13, 13, 14,54, 121,227precosymplectic Lagrangean system 210presymplectic- form 28- Hamilton equations 212- Lagrangean system 210- manifold 28- structure 28primary- constraints 134, 218- constraint set 94, 13I- dynamical constraints 134- semispray-constraint set 94- semis pray constraints 134principle of least time IIprojectable- differential form 33- vector field 31projection 29- of a differential form 33- of a vector field 31prolonged extremal 129proper dynamics 87,129,132,133,217quantization 63, 74, 82, 88, 10I, 125quantum mechanics 63quasi-configuration space 187, 189quasi-holonomic Hamilton extremal 194,
199rank of a distribution 21reducibility condition 73regular- dynamical form I 17- mechanical system 117,220- even-order Lagrangean system 100, 104,
105,105- first-order Lagrangean system 101, 102,
220- first-order Lagrangian 100- Hamiltoninan system 100
Index
- Lagrangean system 95, 97, 98, 100, 129,158
- Lagrangian 97, 98, 223- minimal-order Lagrangian 102- odd-order Lagrangean system 103, 103,
105,183,195- second-order Lagrangean system 102- second-order Lagrangian 101- zero-order Lagrangean system 102, 105,
109, 127regularity 81, 81, 97,129,130regularity condition 17,81,99, 103,213- for Lagrangians 100, 102, 213Riemannian structure 224s-equivalent- Lagrangians 123- mappings 31- sections 29s-jet 29s-jet prolongation- of a fibered manifold 29- of a projectable vector field 32- of a section 30second-order differential equation problem
134second variation 2secondary- constraints 134- dynamical constraints 133, 134- semispray constraints 133, 134section of a fibered manifold 29semi-finslerian manifold 221semi-finslerian structure 223semiregular- Lagrangean system 130, 131, 141, 145,
172,181,183,210- Lagrangian 131semiregularity 129,141- along a submanifold 141semispray 35, 37, 99, 143- along a submanifold 95- connection 37, 39, 98, 193- connection related with a dynamical
form 117- distribution 39set of admissible initial conditions- for Euler-Lagrange equations 134- for Hamilton equations 133singular Lagrangean system 129, 144smooth distribution 21
SODE problem 134soldering form 38, 226solution-equivalent Lagrangians 123source of an s-jet 31Souriau-Lagrange form 60space- rotation 159- translation 159strongly semiregular Lagrangean system
132,143structure- of extremals 90, 190- of second-order Euler-Lagrange
expressions 118support of a deformation 50symmetry- of a differential form 26- of a distribution 24symplectic- form 28, 213- Hamilton equations 213- Lagrangean system 99, 212- manifold 28- reduction 150- structure 28tangent- bundle 31- manifold 31target of an s -jet 31Theorem- on canonical form of a Lepagean
two-form 66- on order-reduction 68- on primary dynamical constraints 94- on primary semispray constraints 94time-dependent- finslerian structure 224- riemannian structure 224time-independent- Hamiltonian 71, 210- Lagrangean system 71, 210time translation 159,209total- energy 12, 71, 210- space a fibered manifold 29trivial Lagrangian 10, 60Vainberg-Tonti Lagrangian 5, 56, 118,
210Van Hove theorem 191variation of a section 50
Index 251
variational- bicomplex 64- connection 11, 123,227- dynamical form 53- function 42- integral 1- integrating factor 9, 123- metric 220- multiplier 9, 123- semispray connection 123, 227- semispray distribution 123- sequence 64variationality conditions 5, 8, 13, 14,57,61,64,118,161
vertical- distribution 32, 40- vector field 31wave front 186weak regularity 129weakly horizontal distribution 40,91weakly regular- autonomous Lagrangean system 217- Lagrangean system 130, 132, 143, 145- Lagrangian 132weakly variational dynamical form 126Weber's electrodynamic force 15zero-order Lagrangean system 73, 75, 83,
88, 90, 100, 101, 102, 109, 127O-contact form 35r -prolongation of a vector field 38tl-prolongation 40
