T

™

m(r2 + r 2 θ2 ) ’ V (r) dt.

S= ™ (1.56)

2

0

Noether observes that the integrand is left unchanged if we make the variation

θ(t) ’ θ(t) + ± (1.57)

1.3. LAGRANGIAN MECHANICS 15

where ± is a ¬xed angle and is a small, time-independent, parameter. This

invariance is the symmetry we shall exploit. It is a mathematical identity:

it does not require that r and θ obey the equations of motion. She next

observes that since the equations of motion are equivalent to the statement

that S is left stationary under any in¬nitesimal variations in r and θ, they

necessarily imply that S is stationary under the speci¬c variation

θ(t) ’ θ(t) + (t)± (1.58)

where now is allowed to be time-dependent. This stationarity of the action

is no longer a mathematical identity, but, because it requires r, θ, to obey

the equations of motion, has physical content. Inserting δθ = (t)± into our

expression for S gives

T

™

r 2 θ ™ dt.

δS = ± (1.59)

0

Note that this variation depends only on the time derivative of , and not

itself. This is because of the invariance of S under time-independent rota-

tions. We now assume that (t) = 0 at t = 0 and t = T , and integrate by

parts to take the time derivative o¬ and put it on the rest of the integrand:

d 2™

δS = ’± (r θ) (t) dt. (1.60)

dt

Since the equations of motion say that δS = 0 under all in¬nitesimal varia-

tions, and in particular those due to any time dependent rotation (t)±, we

deduce that the equations of motion imply that the coe¬cient of (t) must

be zero, and so, provided r(t), θ(t), obey the equations of motion, we have

d 2™

0= (r θ). (1.61)

dt

As a second illustration we derive energy (¬rst integral) conservation for

the case that the system is invariant under time translations ” meaning

that L does not depend explicitly on time. In this case the action integral

is invariant under constant time shifts t ’ t + in the argument of the

dynamical variable:

q(t) ’ q(t + ) ≈ q(t) + q.

™ (1.62)

The equations of motion tell us that that the action will be stationary under

the variation

δq(t) = (t)q,™ (1.63)

16 CHAPTER 1. CALCULUS OF VARIATIONS

where again we now permit the parameter to depend on t. We insert this

variation into

T

S= L dt (1.64)

0

and ¬nd

‚L ‚L

T

δS = q+

™ (¨ + q ™) dt.

q ™ (1.65)

‚q ‚q

™

0

This expression contains undotted ™s. Because of this the change in S is not

obviously zero when is time independent ” but the absence of any explicit

t dependence in L tells us that

dL ‚L ‚L

= q+

™ q.

¨ (1.66)

dt ‚q ‚q

™

As a consequence, for time independent , we have

dL

T

dt = [L]T ,

δS = (1.67)

0

dt

0

showing that the change in S comes entirely from the endpoints of the time

interval. These ¬xed endpoints explicitly break time-translation invariance,

but in a trivial manner. For general (t) we have

dL ‚L

T

δS = (t) + q™

™ dt. (1.68)

dt ‚q

™

0

This equation is an identity. It does not rely on q obeying the equation of

motion. After an integration by parts, taking (t) to be zero at t = 0, T , it

is equivalent to

d ‚L

T

L’

δS = (t) q dt.

™ (1.69)

dt ‚q™

0

Now we assume that q(t) does obey the equations of motion. The variation

principle then says that δS = 0 for any (t), and we deduce that for q(t)

satisfying the equations of motion we have

d ‚L

L’ q = 0.

™ (1.70)

dt ‚q

™

The general strategy that constitutes “Noether™s theorem” must now be

obvious: we look for an invariance of the action under a symmetry trans-

formation with a time-independent parameter. We then observe that if the

1.3. LAGRANGIAN MECHANICS 17

dynamical variables obey the equations of motion, then the action principle

tells us that the action will remain stationary under such a variation of the

dynamical variables even after the parameter is promoted to being time de-

pendent. The resultant variation of S can only depend on time derivatives of

the parameter. We integrate by parts so as to take all the time derivatives o¬

it, and on to the rest of the integrand. Since the parameter is arbitrary, we

deduce that the equations of motion tell us that that its coe¬cient in the in-

tegral must be zero. Since this coe¬cient is the time derivative of something,

this something is conserved.

1.3.3 Many Degrees of Freedom

The extension of the action principle to many degrees of freedom is straight-

forward. As an example consider the small oscillations about equilibrium of

a system with N degrees of freedom. We parametrize the system in terms of

deviations from the equilibrium position and expand out to quadratic order.

We obtain a Lagrangian

N

1 1

Mij q i q j ’ Vij q i q j ,

L= ™™ (1.71)

2 2

i,j=1

where Mij and Vij are N — N symmetric matrices encoding the inertial and

potential energy properties of the system. Now we have one equation

N