lagrange and hamilton

Chapter 2

Lagranges and HamiltonsEquations

In this chapter, we consider two reformulations of Newtonian mechanics, theLagrangian and the Hamiltonian formalism. The rst is naturally associatedwith conguration space, extended by time, while the latter is the naturaldescription for working in phase space.

Lagrange developed his approach in 1764 in a study of the libration ofthe moon, but it is best thought of as a general method of treating dynamicsin terms of generalized coordinates for conguration space. It so transcendsits origin that the Lagrangian is considered the fundamental object whichdescribes a quantum eld theory.

Hamiltons approach arose in 1835 in his unication of the language ofoptics and mechanics. It too had a usefulness far beyond its origin, andthe Hamiltonian is now most familiar as the operator in quantum mechanicswhich determines the evolution in time of the wave function.

We begin by deriving Lagranges equation as a simple change of coordi-nates in an unconstrained system, one which is evolving according to New-tons laws with force laws given by some potential. Lagrangian mechanicsis also and especially useful in the presence of constraints, so we will thenextend the formalism to this more general situation.

35

36 CHAPTER 2. LAGRANGES AND HAMILTONS EQUATIONS

2.1 Lagrangian for unconstrained systems

For a collection of particles with conservative forces described by a potential,we have in inertial cartesian coordinates

mxi = Fi:

The left hand side of this equation is determined by the kinetic energy func-tion as the time derivative of the momentum pi = @T=@ _xi, while the righthand side is a derivative of the potential energy, @U=@xi. As T is indepen-dent of xi and U is independent of _xi in these coordinates, we can write bothsides in terms of the Lagrangian L = T U , which is then a function ofboth the coordinates and their velocities. Thus we have established

d

dt

@L

@ _xi @L@xi

= 0;

which, once we generalize it to arbitrary coordinates, will be known as La-granges equation. Note that we are treating L as a function of the 2Nindependent variables xi and _xi, so that @L=@ _xi means vary one _xi holdingall the other _xj and all the xk xed. Making this particular combination

of T ( _~r) with U(~r) to get the more complicated L(~r; _~r) seems an articialconstruction for the inertial cartesian coordinates, but it has the advantageof preserving the form of Lagranges equations for any set of generalizedcoordinates.

As we did in section 1.3.3, we assume we have a set of generalized coor-dinates fqjg which parameterize all of coordinate space, so that each pointmay be described by the fqjg or by the fxig, i; j 2 [1; N ], and thus each setmay be thought of as a function of the other, and time:

qj = qj(x1; :::xN ; t) xi = xi(q1; :::qN ; t): (2.1)

We may consider L as a function1 of the generalized coordinates qj and _qj,

1Of course we are not saying that L(x; _x; t) is the same function of its coordinates asL(q; _q; t), but rather that these are two functions which agree at the corresponding physicalpoints. More precisely, we are dening a new function ~L(q; _q; t) = L(x(q; t); _x(q; _q; t); t),but we are being physicists and neglecting the tilde. We are treating the Lagrangian hereas a scalar under coordinate transformations, in the sense used in general relativity, thatits value at a given physical point is unchanged by changing the coordinate system usedto dene that point.

2.1. LAGRANGIAN FOR UNCONSTRAINED SYSTEMS 37

and ask whether the same expression in these coordinates

d

dt

@L

@ _qj @L@qj

also vanishes. The chain rule tells us

@L

@ _xj=Xk

@L

@qk

@qk@ _xj

+Xk

@L

@ _qk

@ _qk@ _xj

: (2.2)

The rst term vanishes because qk depends only on the coordinates xk andt, but not on the _xk. From the inverse relation to (1.10),

_qj =Xi

@qj@xi

_xi +@qj@t; (2.3)

we have@ _qj@ _xi

=@qj@xi

:

Using this in (2.2),@L

@ _xi=Xj

@L

@ _qj

@qj@xi

: (2.4)

Lagranges equation involves the time derivative of this. Here what ismeant is not a partial derivative @=@t, holding the point in congurationspace xed, but rather the derivative along the path which the system takes asit moves through conguration space. It is called the stream derivative, aname which comes from fluid mechanics, where it gives the rate at which someproperty dened throughout the fluid, f(~r; t), changes for a xed element offluid as the fluid as a whole flows. We write it as a total derivative to indicatethat we are following the motion rather than evaluating the rate of changeat a xed point in space, as the partial derivative does.

For any function f(x; t) of extended conguration space, this total timederivative is

df

dt=Xj

@f

@xj_xj +

@f

@t: (2.5)

Using Leibnitz rule on (2.4) and using (2.5) in the second term, we nd

d

dt

@L

@ _xi=Xj

d

dt

@L

@ _qj

!@qj@xi

+Xj

@L

@ _qj

Xk

@2qj@xi@xk

_xk +@2qj@xi@t

!: (2.6)


On the other hand, the chain rule also tells us

@L

@xi=Xj

@L

@qj

@qj@xi

+Xj

@L

@ _qj

@ _qj@xi

;

where the last term does not necessarily vanish, as _qj in general depends onboth the coordinates and velocities. In fact, from 2.3,

@ _qj@xi

=Xk

@2qj@xi@xk

_xk +@2qj@xi@t

;

so@L

@xi=Xj

@L

@qj

@qj@xi

+Xj

@L

@ _qj

Xk

@2qj@xi@xk

_xk +@2qj@xi@t

!: (2.7)

Lagranges equation in cartesian coordinates says (2.6) and (2.7) are equal,and in subtracting them the second terms cancel2, so

0 =Xj

d

dt

@L

@ _qj @L@qj

!@qj@xi

:

The matrix @qj=@xi is nonsingular, as it has @xi=@qj as its inverse, so wehave derived Lagranges Equation in generalized coordinates:

d

dt

@L

@ _qj @L@qj

= 0:

Thus we see that Lagranges equations are form invariant under changes ofthe generalized coordinates used to describe the conguration of the system.It is primarily for this reason that this particular and peculiar combinationof kinetic and potential energy is useful. Note that we implicity assume theLagrangian itself transformed like a scalar, in that its value at a given phys-ical point of conguration space is independent of the choice of generalizedcoordinates that describe the point. The change of coordinates itself (2.1) iscalled a point transformation.

2This is why we chose the particular combination we did for the Lagrangian, ratherthan L = T U for some 6= 1. Had we done so, Lagranges equation in cartesiancoordinates would have been d(@L=@ _xj)=dt @L=@xj = 0, and in the subtraction of(2.7) from (2.6), the terms proportional to @L=@ _qi (without a time derivative) wouldnot have cancelled.

2.2. LAGRANGIAN FOR CONSTRAINED SYSTEMS 39

2.2 Lagrangian for Constrained Systems

We now wish to generalize our discussion to include contraints. At the sametime we will also consider possibly nonconservative forces. As we mentionedin section 1.3.2, we often have a system with internal forces whose eect isbetter understood than the forces themselves, with which we may not beconcerned. We will assume the constraints are holonomic, expressible as kreal functions (~r1; :::; ~rn; t) = 0, which are somehow enforced by constraint

forces ~FCi on the particles fig. There may also be other forces, which wewill call FDi and will treat as having a dynamical eect. These are given byknown functions of the conguration and time, possibly but not necessarilyin terms of a potential.

This distinction will seem articial without examples, so it would be wellto keep these two in mind. In each of these cases the full congurationspace is R3, but the constraints restrict the motion to an allowed subspaceof extended conguration space.

1. In section 1.3.2 we discussed a mass on a light rigid rod, the other endof which is xed at the origin. Thus the mass is constrained to havej~rj = L, and the allowed subspace of conguration space is the surfaceof a sphere, independent of time. The rod exerts the constraint forceto avoid compression or expansion. The natural assumption to make isthat the force is in the radial direction, and therefore has no componentin the direction of allowed motions, the tangential directions. That is,for all allowed displacements, ~r, we have ~FC ~r = 0, and the constraintforce does no work.

2. Consider a bead free to slide without friction on the spoke of a rotatingbicycle wheel3, rotating about a xed axis at xed angular velocity !.That is, for the polar angle of inertial coordinates, := !t = 0 isa constraint4, but the r coordinate is unconstrained. Here the allowedsubspace is not time independent, but is a helical sort of structure inextended conguration space. We expect the force exerted by the spokeon the bead to be in the e^ direction. This is again perpendicular toany virtual displacement, by which we mean an allowed change in

3Unlike a real bicycle wheel, we are assuming here that the spoke is directly along aradius of the circle, pointing directly to the axle.

4There is also a constraint z = 0.


conguration at a xed time. It is important to distinguish this virtualdisplacement from a small segment of the trajectory of the particle. Inthis case a virtual displacement is a change in r without a change in ,and is perpendicular to e^. So again, we have the \net virtual work" ofthe constraint forces is zero. It is important to note that this does notmean that the net real work is zero. In a small time interval, the dis-placement ~r includes a component r!t in the tangential direction,and the force of constraint does do work!

We will assume that the constraint forces in general satisfy this restrictionthat no net virtual work is done by the forces of constraint for any possiblevirtual displacement. Newtons law tells us that _~pi = Fi = F

Ci + F

Di . We

can multiply by an arbitrary virtual displacementXi

~FDi _~pi

~ri =

Xi

~FCi ~ri = 0;

where the rst equality would be true even if ~ri did not satisfy the con-straints, but the second requires ~ri to be an allowed virtual displacement.Thus X

i

~FDi _~pi

~ri = 0; (2.8)

which is known as DAlemberts Principle. This gives an equation whichdetermines the motion on the constrained subspace and does not involve theunspecied forces of constraint FC . We drop the superscript D from now on.

Suppose we know generalized coordinates q1; : : : ; qN which parameterizethe constrained subspace, which means ~ri = ~ri(q1; : : : ; qN ; t), for i = 1; : : : ; n,are known functions and the N qs are independent. There are N = 3n k of these independent coordinates, where k is the number of holonomicconstraints. Then @~ri=@qj is no longer an invertable, or even square, matrix,but we still have

~ri =Xj

@~ri@qj

qj +@~ri@t

t:

For the velocity of the particle, divide this by t, giving

~vi =Xj

@~ri@qj

_qj +@~ri@t; (2.9)

but for a virtual displacement t = 0 we have

~ri =Xj

@~ri@qj

qj:


Dierentiating (2.9) we note that,

@~vi@ _qj

=@~ri@qj

; (2.10)

and also@~vi@qj

=Xk

@2~ri@qj@qk

_qk +@2~ri@qj@t

=d

dt

@~ri@qj

; (2.11)

where the last equality comes from applying (2.5), with coordinates qj ratherthan xj, to f = @~ri=@qj. The rst term in the equation (2.8) statingDAlemberts principle is

Xi

~Fi ~ri =Xj

Xi

~Fi @~ri@qj

qj =Xj

Qj qj:

The generalized force Qj has the same form as in the unconstrained case, asgiven by (1.9), but there are only as many of them as there are unconstraineddegrees of freedom.

The second term of (2.8) involves

Xi

_~pi ~ri =Xi

dpidt

@~ri@qj

qj

=Xj

d

dt

Xi

~pi @~ri@qj

!qj

Xij

pi d

dt

@~ri@qj

!qj

=Xj

d

dt

Xi

~pi @~vi@ _qj

!qj

Xij

pi @~vi@qj

qj

=Xj

"d

dt

Xi

mi~vi @~vi@ _qj

Xi

mivi @~vi@qj

#qj

=Xj

"d

dt

@T

@ _qj @T@qj

#qj;

where we used (2.10) and (2.11) to get the third line. Plugging in the ex-pressions we have found for the two terms in DAlemberts Principle,

Xj

"d

dt

@T

@ _qj @T@qj

Qj#qj = 0:


We assumed we had a holonomic system and the qs were all independent,so this equation holds for arbitrary virtual displacements qj, and therefore

d

dt

@T

@ _qj @T@qj

Qj = 0: (2.12)

Now let us restrict ourselves to forces given by a potential, with ~Fi =~riU(f~rg; t), or

Qj = Xi

@~ri@qj

~riU = @~U(fqg; t)@qj

t

:

Notice that Qj depends only on the value of U on the constrained surface.Also, U is independent of the _qis, so

d

dt

@T

@ _qj @T@qj

+@U

@qj= 0 =

d

dt

@(T U)@ _qj

@(T U)@qj

;

or

d

dt

@L

@ _qj @L@qj

= 0: (2.13)

This is Lagranges equation, which we have now derived in the more generalcontext of constrained systems.

2.2.1 Some examples of the use of Lagrangians

Atwoods machine

Atwoods machine consists of two blocks of mass m1 and m2 attached by aninextensible cord which suspends them from a pulley of moment of inertia Iwith frictionless bearings. The kinetic energy is

T =1

2m1 _x

2 +1

2m2 _x

2 +1

2I!2

U = m1gx+m2g(K x) = (m1 m2)gx+ constwhere we have used the xed length of the cord to conclude that the sum ofthe heights of the masses is a constant K. We assume the cord does not slipon the pulley, so the angular velocity of the pulley is ! = _x=r, and

L =1

2(m1 +m2 + I=r

2) _x2 + (m2 m1)gx;


and Lagranges equation gives

d

dt

@L

@ _x @L@x

= 0 = (m1 +m2 + I=r2)x (m2 m1)g:

Notice that we set up our system in terms of only one degree of freedom, theheight of the rst mass. This one degree of freedom parameterizes the linewhich is the allowed subspace of the unconstrained conguration space, athree dimensional space which also has directions corresponding to the angleof the pulley and the height of the second mass. The constraints restrictthese three variables because the string has a xed length and does not slipon the pulley. Note that this formalism has permitted us to solve the problemwithout solving for the forces of constraint, which in this case are the tensionsin the cord on either side of the pulley.

Bead on spoke of wheel

As a second example, reconsider the bead on the spoke of a rotating bicyclewheel. In section (1.3.4) we saw that the kinetic energy is T = 1

2m _r2 +

12mr2!2. If there are no forces other than the constraint forces, U(r; ) 0,

and the Lagrangian is

L =1

2m _r2 +

1

2mr2!2:

The equation of motion for the one degree of freedom is easy enough:

d

dt

@L

@ _r= mr =

@L

@r= mr!2;

which looks like a harmonic oscillator with a negative spring constant, so thesolution is a real exponential instead of oscillating,

r(t) = Ae!t +Be!t:

The velocity-independent term in T acts just like a potential would, and canin fact be considered the potential for the centrifugal force. But we see thatthe total energy T is not conserved but blows up as t!1, T mB2!2e2!t.This is because the force of constraint, while it does no virtual work, does doreal work.


Mass on end of gimballed rod

Finally, let us consider the mass on the end of the gimballed rod. Theallowed subspace is the surface of a sphere, which can be parameterized byan azimuthal angle and the polar angle with the upwards direction, , interms of which

z = cos ; x = sin cos; y = sin sin;

and T = 12m2( _2 + sin2 _2). With an arbitrary potential U(; ), the La-

grangian becomes

L =1

2m2( _2 + sin2 _2) U(; ):

From the two independent variables ; there are two Lagrange equations ofmotion,

m2 = @U@

+1

2sin(2) _2; (2.14)

d

dt

m2 sin2 _

= @U

@: (2.15)

Notice that this is a dynamical system with two coordinates, similar to ordi-nary mechanics in two dimensions, except that the mass matrix, while diag-onal, is coordinate dependent, and the space on which motion occurs is notan innite flat plane, but a curved two dimensional surface, that of a sphere.These two distinctions are connected|the coordinates enter the mass ma-trix because it is impossible to describe a curved space with unconstrainedcartesian coordinates.

Often the potential U(; ) will not actually depend on , in which caseEq. 2.15 tells us m2 sin2 _ is constant in time. We will discuss this furtherin Section 2.4.1.

2.3 Hamiltons Principle

The conguration of a system at any moment is specied by the value of thegeneralized coordinates qj(t), and the space coordinatized by these q1; : : : ; qNis the conguration space. The time evolution of the system is given by

2.3. HAMILTONS PRINCIPLE 45

the trajectory, or motion of the point in conguration space as a function oftime, which can be specied by the functions qi(t).

One can imagine the system taking many paths, whether they obey New-tons Laws or not. We consider only paths for which the qi(t) are dieren-tiable. Along any such path, we dene the action as

S =Z t2t1L(q(t); _q(t); t)dt: (2.16)

The action depends on the starting and ending points q(t1) and q(t2), butbeyond that, the value of the action depends on the path, unlike the workdone by a conservative force on a point moving in ordinary space. In fact,it is exactly this dependence on the path which makes this concept useful| Hamiltons principle states that the actual motion of the particle fromq(t1) = qi to q(t2) = qf is along a path q(t) for which the action is stationary.That means that for any small deviation of the path from the actual one,keeping the initial and nal congurations xed, the variation of the actionvanishes to rst order in the deviation.

To nd out where a dierentiable function of one variable has a stationarypoint, we dierentiate and solve the equation found by setting the derivativeto zero. If we have a dierentiable function f of several variables xi, therst-order variation of the function is f =

Pi(xix0i) @f=@xijx0 , so unless

@f=@xijx0 = 0 for all i, there is some variation of the fxig which causes arst order variation of f , and then x0 is not a stationary point.

But our action is a functional, a function of functions, which representan innite number of variables, even for a path in only one dimension. In-tuitively, at each time q(t) is a separate variable, though varying q at onlyone point makes _q hard to interpret. A rigorous mathematician might wantto describe the path q(t) on t 2 [0; 1] in terms of Fourier series, for whichq(t) = q0 + q1t+

Pn=1 an sin(nt). Then the functional S(f) given by

S =Zf(q(t); _q(t); t)dt

becomes a function of the innitely many variables q0; q1; a1; : : :. The end-points x q0 and q1, but the stationary condition gives an innite number ofequations @S=@an = 0.

It is not really necessary to be so rigorous, however. Under a changeq(t) ! q(t) + q(t), the derivative will vary by _q = d q(t)=dt, and the


functional S will vary by

S =Z @f

@qq +

@f

@ _q _q

!dt

=@f

@ _qq

f

i

+Z "@f

@q ddt

@f

@ _q

#qdt;

where we integrated the second term by parts. The boundary terms each havea factor of q at the initial or nal point, which vanish because Hamilton tellsus to hold the qi and qf xed, and therefore the functional is stationary ifand only if

@f

@q ddt

@f

@ _q= 0 for t 2 (ti; tf ) (2.17)

We see that if f is the Lagrangian, we get exactly Lagranges equation. Theabove derivation is essentially unaltered if we have many degrees of freedomqi instead of just one.

2.3.1 Examples of functional variation

In this section we will work through some examples of functional variationsboth in the context of the action and for other examples not directly relatedto mechanics.

The falling particle

As a rst example of functional variation, consider a particle thrown up ina uniform gravitional eld at t = 0, which lands at the same spot at t = T .The Lagrangian is L = 1

2m( _x2 + _y2 + _z2)mgz, and the boundary conditions

are x(t) = y(t) = z(t) = 0 at t = 0 and t = T . Elementary mechanics tellsus the solution to this problem is x(t) = y(t) 0, z(t) = v0t 12gt2 withv0 =

12gT . Let us evaluate the action for any other path, writing z(t) in

terms of its deviation from the suspected solution,

z(t) = z(t) +1

2gT t 1

2gt2:

We make no assumptions about this path other than that it is dierentiableand meets the boundary conditions x = y = z = 0 at t = 0 and at t = T .

2.3. HAMILTONS PRINCIPLE 47

The action is

S =Z T0

(1

2m

24 _x2 + _y2 + dzdt

!2+ g(T 2t)dz

dt+

1

4g2(T 2t)2

35mgz 1

2mg2t(T t)

)dt:

The fourth term can be integrated by parts,

Z T0

1

2mg(T 2t)dz

dtdt =

1

2mg(T 2t)z

T0

+Z T0mgz(t) dt:

The boundary term vanishes because z = 0 where it is evaluated, and theother term cancels the sixth term in S, so

S =Z T0

1

2mg2

1

4(T 2t)2 t(T t)

dt

+Z T0

1

2m

24 _x2 + _y2 + dzdt

!235 :The rst integral is independent of the path, so the minimum action requiresthe second integral to be as small as possible. But it is an integral of a non-negative quantity, so its minimum is zero, requiring _x = _y = dz=dt = 0.As x = y = z = 0 at t = 0, this tells us x = y = z = 0 at all times, andthe path which minimizes the action is the one we expect from elementarymechanics.

Is the shortest path a straight line?

The calculus of variations occurs in other contexts, some of which are moreintuitive. The classic example is to nd the shortest path between two pointsin the plane. The length of a path y(x) from (x1; y1) to (x2; y2) is given

5 by

=Z x2x1

ds =Z x2x1

vuut1 + dydx

!2dx:

5Here we are assuming the path is monotone in x, without moving somewhere to theleft and somewhere to the right. To prove that the straight line is shorter than other pathswhich might not obey this restriction, do Exercise 2.2.


We see that length is playing the role of the action, and x is playing the roleof t. Using _y to represent dy=dx, we have the integrand f(y; _y; x) =

p1 + _y2,

and @f=@y = 0, so Eq. 2.17 gives

d

dx

@f

@ _y=

d

dx

_yp1 + _y2

= 0; so _y = const.

and the path is a straight line.

2.4 Conserved Quantities

2.4.1 Ignorable Coordinates

If the Lagrangian does not depend on one coordinate, say qk, then we sayit is an ignorable coordinate. Of course, we still want to solve for it, asits derivative may still enter the Lagrangian and eect the evolution of othercoordinates. By Lagranges equation

d

dt

@L

@ _qk=@L

@qk= 0;

so if in general we dene

Pk :=@L

@ _qk;

as the generalized momentum, then in the case that L is independent ofqk, Pk is conserved, dPk=dt = 0.

Linear Momentum

As a very elementary example, consider a particle under a force given by apotential which depends only on y and z, but not x. Then

L =1

2m

_x2 + _y2 + _z2 U(y; z)

is independent of x, x is an ignorable coordinate and

Px =@L

@ _x= m _x

is conserved. This is no surprize, of course, because the force is F = rUand Fx = @U=@x = 0.

2.4. CONSERVED QUANTITIES 49

Note that, using the denition of the generalized momenta

Pk =@L

@ _qk;

Lagranges equation can be written as

d

dtPk =

@L

@qk=@T

@qk @U@qk

:

Only the last term enters the denition of the generalized force, so if thekinetic energy depends on the coordinates, as will often be the case, it isnot true that dPk=dt = Qk. In that sense we might say that the generalizedmomentum and the generalized force have not been dened consistently.

Angular Momentum

As a second example of a system with an ignorable coordinate, consider anaxially symmetric system described with inertial polar coordinates (r; ; z),with z along the symmetry axis. Extending the form of the kinetic energywe found in sec (1.3.4) to include the z coordinate, we have T = 1

2m _r2 +

12mr2 _2 + 1

2m _z2. The potential is independent of , because otherwise the

system would not be symmetric about the z-axis, so the Lagrangian

L =1

2m _r2 +

1

2mr2 _2 +

1

2m _z2 U(r; z)

does not depend on , which is therefore an ignorable coordinate, and

P :=@L

@ _= mr2 _ = constant:

We see that the conserved momentum P is in fact the z-component of theangular momentum, and is conserved because the axially symmetric potentialcan exert no torque in the z-direction:

z = ~r ~rU

z

= r~rU

= r2@U@

= 0:

Finally, consider a particle in a spherically symmetric potential in spher-ical coordinates. In section (3.1.2) we will show that the kinetic energy in


spherical coordinates is T = 12m _r2 + 1

2mr2 _2 + 1

2mr2 sin2 _2, so the La-

grangian with a spherically symmetric potential is

L =1

2m _r2 +

1

2mr2 _2 +

1

2mr2 sin2 _2 U(r):

Again, is an ignorable coordinate and the conjugate momentum P isconserved. Note, however, that even though the potential is independent of as well, does appear undierentiated in the Lagrangian, and it is not anignorable coordinate, nor is P conserved

6.If qj is an ignorable coordinate, not appearing undierentiated in the

Lagrangian, any possible motion qj(t) is related to a dierent trajectoryq0j(t) = qj(t) + c, in the sense that they have the same action, and if oneis an extremal path, so will the other be. Thus there is a symmetry of thesystem under qj ! qj + c, a continuous symmetry in the sense that c cantake on any value. As we shall see in Section 8.3, such symmetries generallylead to conserved quantities. The symmetries can be less transparent thanan ignorable coordinate, however, as in the case just considered, of angularmomentum for a spherically symmetric potential, in which the conservationof Lz follows from an ignorable coordinate , but the conservation of Lx andLy follow from symmetry under rotation about the x and y axes respectively,and these are less apparent in the form of the Lagrangian.

2.4.2 Energy Conservation

We may ask what happens to the Lagrangian along the path of the motion.

dL

dt=

Xi

@L

@qi

dqidt

+Xi

@L

@ _qi

d _qidt

+@L

@t

In the rst term the rst factor is

d

dt

@L

@ _qi

6It seems curious that we are nding straightforwardly one of the components of theconserved momentum, but not the other two, Ly and Lx, which are also conserved. Thefact that not all of these emerge as conjugates to ignorable coordinates is related to the factthat the components of the angular momentum do not commute in quantum mechanics.This will be discussed further in section (6.6.1).

2.4. CONSERVED QUANTITIES 51

by the equations of motion, so

dL

dt=

d

dt

Xi

@L

@ _qi_qi

!+@L

@t:

We expect energy conservation when the potential is time invariant and thereis not time dependence in the constraints, i.e. when @L=@t = 0, so we rewritethis in terms of

H(q; _q; t) =Xi

_qi@L

@ _qi L = X

i

_qiPi L

Then for the actual motion of the system,

dH

dt= @L

@t:

If @L=@t = 0, H is conserved.H is essentially the Hamiltonian, although strictly speaking that name

is reserved for the function H(q; p; t) on extended phase space rather thanthe function with arguments (q; _q; t). What is H physically? In the caseof Newtonian mechanics with a potential function, L is an inhomogeneousquadratic function of the velocities _qi. If we write the Lagrangian L = L2 +L1 + L0 as a sum of pieces purely quadratic, purely linear, and independentof the velocities respectively, then

Xi

_qi@

@ _qi

is an operator which multiplies each term by its order in velocities,

Xi

_qi@Ln@ _qi

= nLn;Xi

_qi@L

@ _qi= 2L2 + L1;

andH = L2 L0:

For a system of particles described by their cartesian coordinates, L2 isjust the kinetic energy T , while L0 is the negative of the potential energyL0 = U , so H = T + U is the ordinary energy. There are, however, con-strained systems, such as the bead on a spoke of Section 2.2.1, for which theHamiltonian is conserved but is not the ordinary energy.


2.5 Hamiltons Equations

We have written the Lagrangian as a function of qi, _qi, and t, so it is afunction of N +N + 1 variables. For a free particle we can write the kineticenergy either as 1

2m _x2 or as p2=2m. More generally, we can7 reexpress the

dynamics in terms of the 2N + 1 variables qk, Pk, and t.The motion of the system sweeps out a path in the space (q; _q; t) or a

path in (q; P; t). Along this line, the variation of L is

dL =Xk

@L

@ _qkd _qk +

@L

@qkdqk

!+@L

@tdt

=Xk

Pkd _qk + _Pkdqk

+@L

@tdt

where for the rst term we used the denition of the generalized momentumand in the second we have used the equations of motion _Pk = @L=@qk. Thenexamining the change in the Hamiltonian H =

Pk Pk _qkL along this actual

motion,

dH =Xk

(Pkd _qk + _qkdPk) dL

=Xk

_qkdPk _Pkdqk

@L@tdt:

If we think of _qk and H as functions of q and P , and think of H as a functionof q, P , and t, we see that the physical motion obeys

_qk =@H

@Pk

q;t

; _Pk = @H@qk

P;t

;@H

@t

q;P

= @L@t

q; _q

The rst two constitute Hamiltons equations of motion, which are rstorder equations for the motion of the point representing the system in phasespace.

Lets work out a simple example, the one dimensional harmonic oscillator.Here the kinetic energy is T = 1

2m _x2, the potential energy is U = 1

2kx2, so

7In eld theory there arise situations in which the set of functions Pk(qi; _qi) cannot beinverted to give functions _qi = _qi(qj ; Pj). This gives rise to local gauge invariance, andwill be discussed in Chapter 8, but until then we will assume that the phase space (q; p),or cotangent bundle, is equivalent to the tangent bundle, i.e. the space of (q; _q).

2.5. HAMILTONS EQUATIONS 53

L = 12m _x2 1

2kx2, the only generalized momentum is P = @L=@ _x = m _x, and

the Hamiltonian is H = P _xL = P 2=m(P 2=2m 12kx2) = P 2=2m+ 1

2kx2.

Note this is just the sum of the kinetic and potential energies, or the totalenergy.

Hamiltons equations give

_x =@H

@P

x

=P

m; _P = @H

@x

P

= kx = F:

These two equations verify the usual connection of the momentum and ve-locity and give Newtons second law.

The identication of H with the total energy is more general than ourparticular example. If T is purely quadratic in velocities, we can write T =12

PijMij _qi _qj in terms of a symmetric mass matrix Mij. If in addition U is

independent of velocities,

L =1

2

Xij

Mij _qi _qj U(q)

Pk =@L

@ _qk=Xi

Mki _qi

which as a matrix equation in a n-dimensional space is P = M _q. AssumingM is invertible,8 we also have _q = M1 P , so

H = P T _q L= P T M1 P

1

2_qT M _q U(q)

= P T M1 P 1

2P T M1 M M1 P + U(q)

=1

2P T M1 P + U(q) = T + U

so we see that the Hamiltonian is indeed the total energy under these cir-cumstances.

8If M were not invertible, there would be a linear combination of velocities whichdoes not aect the Lagrangian. The degree of freedom corresponding to this combinationwould have a Lagrange equation without time derivatives, so it would be a constraintequation rather than an equation of motion. But we are assuming that the qs are a setof independent generalized coordinates that have already been pruned of all constraints.


2.6 Dont plug Equations of Motion into the

Lagrangian!

When we have a Lagrangian with an ignorable coordinate, say , and there-fore a conjugate momentum P which is conserved and can be considereda constant, we are able to reduce the problem to one involving one fewerdegrees of freedom. That is, one can substitute into the other dierentialequations the value of _ in terms of P and other degrees of freedom, sothat and its derivatives no longer appear in the equations of motion. Forexample, consider the two dimensional isotropic harmonic oscillator,

L =1

2m

_x2 + _y2 1

2kx2 + y2

=

1

2m_r2 + r2 _2

1

2kr2

in polar coordinates. The equations of motion are

_P = 0; where P = mr2 _;

mr = kr +mr _2 =) mr = kr + P 2.mr3:

The last equation is now a problem in the one degree of freedom r.

One might be tempted to substitute for _ into the Lagrangianand then have a Lagrangian involving one fewer degrees of free-dom. In our example, we would get

L =1

2m _r2 +

P 22mr2

12kr2;

which gives the equation of motion

mr = P2

mr3 kr:

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

This iswrong

Notice that the last equation has the sign of the P 2 term reversed fromthe correct equation. Why did we get the wrong answer? In deriving theLagrange equation which comes from varying r, we need

d

dt

@L

@ _r

r;; _

=@L

@r

_r;; _

:

2.7. VELOCITY-DEPENDENT FORCES 55

But we treated P as xed, which means that when we vary r on the righthand side, we are not holding _ xed, as we should be. While we oftenwrite partial derivatives without specifying explicitly what is being held xed,they are not dened without such a specication, which we are expected tounderstand implicitly. However, there are several examples in Physics, suchas thermodynamics, where this implicit understanding can be unclear, andthe results may not be what was intended.

2.7 Velocity-dependent forces

We have concentrated thus far on Newtonian mechanics with a potentialgiven as a function of coordinates only. As the potential is a piece of theLagrangian, which may depend on velocities as well, we should also entertainthe possibility of velocity-dependent potentials. Only by considering such apotential can we possibly nd velocity-dependent forces, and one of the mostimportant force laws in physics is of that form. This is the Lorentz force9

on a particle of charge q in the presence of electromagnetic elds ~E(~r; t) and~B(~r; t),

~F = q

~E +

~v

c ~B

!: (2.18)

If the motion of a charged particle is described by Lagrangian mechanics witha potential U(~r;~v; t), Lagranges equation says

0 =d

dt

@L

@vi @L@ri

= mri ddt

@U

@vi+@U

@ri; so Fi =

d

dt

@U

@vi @U@ri

:

We want a force linear in ~v and proportional to q, so let us try

U = q(~r; t) + ~v ~C(~r; t)

:

Then we need to have

~E +~v

c ~B = d

dt~C ~rX

j

vj ~rCj: (2.19)

9We have used Gaussian units here, but those who prefer S. I. units (rationalized MKS)can simply set c = 1.


The rst term is a stream derivative evaluated at the time-dependent positionof the particle, so, as in Eq. (2.5),

d

dt~C =

@ ~C

@t+Xj

vj@ ~C

@xj:

The last term looks like the last term of (2.19), except that the indices on the

derivative operator and on ~C have been reversed. This suggests that thesetwo terms combine to form a cross product. Indeed, noting (A.17) that

~v ~r ~C

=Xj

vj ~rCj X

vj@ ~C

@xj;

we see that (2.19) becomes

~E+~v

c ~B = @

~C

@t ~rX

j

vj ~rCj +Xj

vj@ ~C

@xj=@ ~C

@t ~r~v

~r ~C

:

We have successfully generated the term linear in ~v if we can show thatthere exists a vector eld ~C(~r; t) such that ~B = c~r ~C. A curl is alwaysdivergenceless, so this requires ~r ~B = 0, but this is indeed one of Maxwellsequations, and it ensures10 there exists a vector eld ~A, known as the mag-netic vector potential, such that ~B = ~r ~A. Thus with ~C = ~A=c, weneed only to nd a such that

~E = ~r 1c

@ ~A

@t:

Once again, one of Maxwells laws,

~r ~E + 1c

@ ~B

@t= 0;

guarantees the existence of , the electrostatic potential, because afterinserting ~B = ~r ~A, this is a statement that ~E + (1=c)@ ~A=@t has no curl,and is the gradient of something.

10This is but one of many consequences of the Poincare lemma, discussed in section 6.5(well, it should be). The particular forms we are using here state that if ~r ~B = 0 and~r ~F = 0 in all of R3, then there exist a scalar function and a vector eld ~A such that~B = ~r ~A and ~F = ~r.


Thus we see that the Lagrangian which describes the motion of a chargedparticle in an electromagnetic eld is given by a velocity-dependent potential

U(~r;~v) = q(r; t) (~v=c) ~A(~r; t)

:

Note, however, that this Lagrangian describes only the motion of the chargedparticle, and not the dynamics of the eld itself.

Arbitrariness in the Lagrangian In this discussion of nding the La-grangian to describe the Lorentz force, we used the lemma that guaranteedthat the divergenceless magnetic eld ~B can be written in terms of somemagnetic vector potential ~A, with ~B = ~r ~A. But ~A is not uniquely spec-ied by ~B; in fact, if a change is made, ~A ! ~A + ~r(~r; t), ~B is unchangedbecause the curl of a gradient vanishes. The electric eld ~E will be changedby (1=c)@ ~A=@t, however, unless we also make a change in the electrostaticpotential, ! (1=c)@=@t. If we do, we have completely unchangedelectromagnetic elds, which is where the physics lies. This change in thepotentials,

~A! ~A+ ~r(~r; t); ! (1=c)@=@t; (2.20)is known as a gauge transformation, and the invariance of the physicsunder this change is known as gauge invariance. Under this change, thepotential U and the Lagrangian are not unchanged,

L! L q ~v

c ~A

!= L+

q

c

@

@t+q

c~v ~r(~r; t) = L+ q

c

d

dt:

We have here an example which points out that there is not a uniqueLagrangian which describes a given physical problem, and the ambiguity ismore that just the arbitrary constant we always knew was involved in thepotential energy. This ambiguity is quite general, not depending on the gaugetransformations of Maxwell elds. In general, if

L(2)(qj; _qj; t) = L(1)(qj; _qj; t) +

d

dtf(qj; t) (2.21)

then L(1) and L(2) give the same equations of motion, and therefore the samephysics, for qj(t). While this can be easily checked by evaluating the Lagrangeequations, it is best understood in terms of the variation of the action. For


any path qj(t) between qjI at t = tI to qjF at t = tF , the two actions arerelated by

S(2) =Z tFtI

L(1)(qj; _qj; t) +

d

dtf(qj; t)

!dt

= S(1) + f(qjF ; tF ) f(qjI ; tI):

The variation of path that one makes to nd the stationary action does notchange the endpoints qjF and qjI , so the dierence S

(2) S(1) is a constantindependent of the trajectory, and a stationary trajectory for S(2) is clearlystationary for S(1) as well.

The conjugate momenta are aected by the change in Lagrangian, how-ever, because L(2) = L(1) +

Pj _qj@f=@qj + @f=@t, so

p(2)j =

@L(2)

@ _qj= p

(1)j +

@f

@qj:

This ambiguity is not usually mentioned in elementary mechanics, be-cause if we restict our attention to Lagrangians consisting of canonical kineticenergy and potentials which are velocity-independent, a change (2.21) to aLagrangian L(1) of this type will produce an L(2) which is not of this type, un-less f is independent of position q and leaves the momenta unchanged. Thatis, the only f which leaves U velocity independent is an arbitrary constant.

Dissipation Another familiar force which is velocity dependent is friction.Even the \constant" sliding friction met with in elementary courses dependson the direction, if not the magnitude, of the velocity. Friction in a viscousmedium is often taken to be a force proportional to the velocity, ~F = ~v.We saw above that a potential linear in velocities produces a force perpen-dicular to ~v, and a term higher order in velocities will contribute a forcethat depends on acceleration. This situation cannot handled by Lagrangesequations. More generally, a Lagrangian can produce a force Qi = Rij _qj withantisymmetric Rij, but not for a symmetric matrix. An extension to the La-grange formalism, involving Rayleighs dissipation function, can handle sucha case. These dissipative forces are discussed in Ref. [6].

Exercises


2.1 (Galelean relativity): Sally is sitting in a railroad car observing a system ofparticles, using a Cartesian coordinate system so that the particles are at positions~r

(S)i (t), and move under the influence of a potential U

(S)(f~r (S)i g). Thomas is inanother railroad car, moving with constant velocity ~u with respect to Sally, and sohe describes the position of each particle as ~r (T )i (t) = ~r

(S)i (t) ~ut. Each takes the

kinetic energy to be of the standard form in his system, i.e. T (S) = 12Pmi

_~r (S)i2

and T (T ) = 12Pmi

_~r (T )i2

.

(a) Show that if Thomas assumes the potential function U (T )(~r (T )) to be the sameas Sallys at the same physical points,

U (T )(~r (T )) = U (S)(~r (T ) + ~ut); (2.22)

then the equations of motion derived by Sally and Thomas describe the samephysics. That is, if r (S)i (t) is a solution of Sallys equations, r

(T )i (t) = r

(S)i (t) ~ut

is a solution of Thomas.(b) show that if U (S) (f~rig) is a function only of the displacements of one particlefrom another, f~ri ~rjg, then U (T ) is the same function of its arguments as U (S),U (T )(f~rig) = U (S)(f~rig). This is a dierent statement than Eq. 2.22, which statesthat they agree at the same physical conguration. Show it will not generally betrue if U (S) is not restricted to depend only on the dierences in positions.(c) If it is true that U (S)(~r) = U (T )(~r), show that Sally and Thomas derive thesame equations of motion, which we call \form invariance" of the equations.(d) Show that nonetheless Sally and Thomas disagree on the energy of a particularphysical motion, and relate the dierence to the total momentum. Which of thesequantities are conserved?

2.2 In order to show that the shortest path in two dimensional Euclidean spaceis a straight line without making the assumption that x does not change signalong the path, we can consider using a parameter and describing the path bytwo functions x() and y(), say with 2 [0; 1]. Then

=Z 10dq

_x2() + _y2();

where _x means dx=d. This is of the form of a variational integral with twovariables. Show that the variational equations do not determine the functionsx() and y(), but do determine that the path is a straight line. Show that thepair of functions (x(); y()) gives the same action as another pair (~x(); ~y()),where ~x() = x(t()) and ~y() = y(t()), where t() is any monotone functionmapping [0; 1] onto itself. Explain why this equality of the lengths is obvious


in terms of alternate parameterizations of the path. [In eld theory, this is anexample of a local gauge invariance, and plays a major role in string theory.]

2.3 Consider a circular hoop of radius R rotating about a vertical diameter ata xed angular velocity . On the hoop there is a bead of mass m, which slideswithout friction on the hoop. The only external force is gravity. Derive theLagrangian and the Lagrange equation using the polar angle as the unconstrainedgeneralized coordinate. Find a conserved quantity, and nd the equilibrium points,for which _ = 0. Find the condition on such that there is an equilibrium pointaway from the axis.

2.4 Early steam engines had a feedback device, called a governor, to automat-ically control the speed. The engine rotated a vertical shaft with an angular

velocity proportional to its speed. On oppo-site sides of this shaft, two hinged rods eachheld a metal weight, which was attached toanother such rod hinged to a sliding collar, asshown.

As the shaft rotates faster, the balls moveoutwards, the collar rises and uncovers a hole,releasing some steam. Assume all hinges arefrictionless, the rods massless, and each ballhas mass m1 and the collar has mass m2.

(a) Write the Lagrangian in terms of the gen-eralized coordinate .

(b) Find the equilibrium angle as a func-tion of the shaft angular velocity . Tellwhether the equilibrium is stable or not.

m1 1

m

L

m2

W

L

Governor for a steam en-gine.

2.5 A transformer consists of two coils of conductor each of which has an induc-tance, but which also have a coupling, or mutual inductance.


If the current flowing into the upper posts of coilsA and B are IA(t) and IB(t) respectively, the volt-age dierence or EMF across each coil is VA and VBrespectively, where

VA = LAdIAdt

+MdIBdt

VB = LBdIBdt

+MdIAdt

A B

V

0

V

0

I IA BA B

Consider the circuit shown, twocapacitors coupled by a such a trans-former, where the capacitances areCA and CB respectively, with thecharges q1(t) and q2(t) serving as thegeneralized coordinates for this prob-lem. Write down the two second or-der dierential equations of \motion"for q1(t) and q2(t), and write a La-grangian for this system.

q

q

qq

1 2A B

1 2

2.6 A cylinder of radius R is held horizontally in a xed position, and a smalleruniform cylindrical disk of radius a is placed on top of the rst cylinder, and isreleased from rest. There is a coecient ofstatic friction s and a coecient of kineticfriction k < s for the contact between thecylinders. As the equilibrium at the top isunstable, the top cylinder will begin to roll onthe bottom cylinder.

(a) If s is suciently large, the small diskwill roll until it separates from the xedcylinder. Find the angle at which theseparation occurs, and nd the mini-mum value of s for which this situationholds.

(b) If s is less than the minimum valuefound above, what happens dierently,and at what angle does this dierentbehavior begin?

q

a

R

A small cylinder rolling ona xed larger cylinder.

2.7 (a) Show that if (q1; :::; qn; t) is an arbitrary dierentiable function on ex-tended conguration space, and L(1)(fqig; f _qjg; t) and L(2)(fqig; f _qjg; t) are two


Lagrangians which dier by the total time derivative of ,

L(1)(fqig; f _qjg; t) = L(2)(fqig; f _qjg; t) + ddt

(q1; :::; qn; t);

show by explicit calculations that the equations of motion determined by L(1) arethe same as the equations of motion determined by L(2).(b) What is the relationship between the momenta p(1)i and p

(2)i determined by

these two Lagrangians respectively.

2.8 A particle of mass m1 moves in two dimensions on a frictionless horizontaltable with a tiny hole in it. An inextensible massless string attached to m1 goesthrough the hole and is connected to another particle of mass m2, which movesvertically only. Give a full set of generalized unconstrained coordinates and writethe Lagrangian in terms of these. Assume the string remains taut at all timesand that the motions in question never have either particle reaching the hole, andthere is no friction of the string sliding at the hole.Are there ignorable coordinates? Reduce the problem to a single second orderdierential equation. Show this is equivalent to single particle motion in onedimension with a potential V (r), and nd V (r).

2.9 Consider a mass m on the end of a massless rigid rod of length , the otherend of which is free to rotate about a xed point. This is a spherical pendulum.Find the Lagrangian and the equations of motion.

2.10 (a) Find a dierential equation for () for the shortest path on the surfaceof a sphere between two arbitrary points on that surface, by minimizing the lengthof the path, assuming it to be monotone in .(b) By geometrical argument (that it must be a great circle) argue that the pathshould satisfy

cos( 0) = K cot ;and show that this is indeed the solution of the dierential equation you derived.

2.11 Consider some intelligent bugs who live on a turntable which, accordingto inertial observers, is spinning at angular velocity ! about its center. At anyone time, the inertial observer can describe the points on the turntable with polarcoordinates r; . If the bugs measure distances between two objects at rest withrespect to them, at innitesimally close points, they will nd


d2 = dr2 +r2

1 !2r2=c2d2;

because their metersticks shrink in thetangential direction and it takes more ofthem to cover the distance we think ofas rd, though their metersticks agreewith ours when measuring radial dis-placements.

The bugs will declare a curve to bea geodesic, or the shortest path betweentwo points, if

Rd is a minimum. Show

that this requires that r() satises

dr

d= r

1 !2r2=c2p2r2 1;

where is a constant.

Straight lines to us and to the bugs,between the same two points.

2.12 Hamiltons Principle tells us that the motion of a particle is determinedby the action functional being stationary under small variations of the path inextended conguration space (t; ~x). The unsymmetrical treatment of t and ~x(t)is not suitable for relativity, but we may still associate an action with each path,which we can parameterize with , so is the trajectory ! (t(); ~x()).In the general relativistic treatment of a particles motion in a gravitational eld,the action is given by mc2 , where is the elapsed proper time, =

Rd .

But distances and time intervals are measured with a spatial varying metric g ,with and ranging from 0 to 3, with the zeroth component referring to time.The four components of extended conguration space are written x, with a su-perscript rather than a subscript, and x0 = ct. The gravitational eld is de-scribed by the space-time dependence of the metric g(x). In this language,an innitesimal element of the path of a particle corresponds to a proper timed = (1=c)

qP gdx

dx , so

S = mc2 = mcZd

vuutX

g(x)dx

d

dx

d:

(a) Find the four Lagrange equations which follow from varying x().

(b) Show that if we multiply these four equations by _x and sum on , we get anidentity rather than a dierential equation helping to determine the functions


x(). Explain this as a consequence of the fact that any path has a lengthunchanged by a reparameterization of the path, ! (), x0() = x(()

(c) Using this freedom to choose to be , the proper time from the start of thepath to the point in question, show that the equations of motion are

d2x

d2+X

dx

d

dx

d= 0;

and nd the expression for .

2.13 (a): Find the canonical momenta for a charged particle moving in an electro-magnetic eld and also under the influence of a non-electromagnetic force describedby a potential U(~r).(b): If the electromagnetic eld is a constant magnetic eld ~B = B0e^z, with noelectric eld and with U(~r) = 0, what conserved quantities are there?

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