‚2L ‚2L

dωL = i j dp § dx + i j dxj § dxi ,

j i

‚p ‚p ‚p ‚x

so

‚2L ‚2L

γ dωL = i j (y, y) y dx ’ y dp + i j (y, y) y j dxi ’ y i dxj .

j i i j

¨ ™¨ ™ ™™ ™

‚p ‚p ‚p ‚x

On the other hand, an easy computation yields

‚2L ‚2L

‚L

’dEL (γ) = (y, y) ’ j i (y, y)y j dx ’ i j (y, y)y i dpj .

i

™ ™ ™™ ™™

i

‚x ‚p ‚x ‚p ‚p

Comparing these last two equations, the condition γ dωL = ’dEL (γ) is seen to be the

¨ ™

Euler-Lagrange equations, as desired.

Conservation of Energy. One important consequence of Proposition 3 is that the

function EL is constant along the curve γ for any L-critical curve γ: [a, b] ’ M. This

™

follows since, for such a curve,

dEL γ (t) = ’dωL γ (t), γ (t) = 0.

¨ ¨ ¨

EL is generally interpreted as the “energy” of the Lagrangian L, and this constancy of EL

on L-critical curves is often called the principle of Conservation of Energy.

Some sources de¬ne EL as L ’ dL(R). My choice was to have EL agree with the

classical energy in the classical problems.

L.4.6 66

De¬nition 3: If L: T M ’ R is a Lagrangian on M, a di¬eomorphism f: M ’ M is said

to be a symmetry of L if L is invariant under the induced di¬eomorphism f : T M ’ T M,

i.e., if L —¦ f = L. A vector ¬eld X on M is said to be an in¬nitesimal symmetry of L if

the (local) ¬‚ow ¦t of X is a symmetry of L for all t.

It is perhaps necessary to make a remark about the last part of this de¬nition. For a

vector ¬eld X which is not necessarily complete, and for any t ∈ R, the “time t” local ¬‚ow

of X is well-de¬ned on an open set Ut ‚ M. The local ¬‚ow of X then gives a well-de¬ned

di¬eomorphism ¦t : Ut ’ U’t . The requirement for X is that, for each t for which Ut = …,

the induced map ¦t : T Ut ’ T U’t should satisfy L —¦ ¦t = L. (Of course, if X is complete,

then Ut = M for all t, so symmetry has its usual meaning.)

Let X be any vector ¬eld on M with local ¬‚ow ¦. This induces a local ¬‚ow on T M

which is associated to a vector ¬eld X on T M. If, in a local coordinate chart, x: U ’ Rn ,

the vector ¬eld X has the expression

‚

X = ai (x) ,

‚xi

then the reader may check that, in the associated canonical coordinates on T U ,

i

‚ j ‚a ‚

i

X =a +p .

‚xi ‚xj ‚pi

The condition that X be an in¬nitesimal symmetry of L is then that L be invariant

under the ¬‚ow of X , i.e., that

i

‚L j ‚a ‚L

i

dL(X ) = a +p = 0.

‚xi ‚xj ‚pi

The following theorem is now a simple calculation. Nevertheless, it is the foundation

of a vast theory. It usually goes by the name “Noether™s Theorem”, though, in fact,

Noether™s Theorem is more general.

Theorem 1: If X is an in¬nitesimal symmetry of the Lagrangian L, then the function

ωL (X ) is constant on γ: [a, b] ’ T M for every L-critical path γ: [a, b] ’ M.

™

Proof: Since the ¬‚ow of X ¬xes L it should not be too surprising that it also ¬xes EL

and ωL . These facts are easily checked by the reader in local coordinates, so they are left

as exercises. In particular,

LX ωL = d(X ωL ) + X dωL = 0 and LX EL = dEL (X ) = 0.

Thus, for any L-critical curve γ in M,

ωL ) γ (t) = ’(X

d ωL (X ) γ (t) = d(X

¨ ¨ dωL ) γ (t)

¨

= dωL γ (t), X (γ(t)) = γ (t) dωL X (γ(t))

¨ ™ ¨ ™

= ’dEL X (γ(t)) = 0.

™

L.4.7 67

Hence, the function ωL (X ) is constant on γ, as desired.

™

Of course, the formula for ωL (X ) in local canonical coordinates is simply

‚L

ωL (X ) = ai ,

‚pi

and the constancy of this function on the solution curves of the Euler-Lagrange equations

is not di¬cult to check directly.

The principle

Symmetry =’ Conservation Law

is so fundamental that whenever a new system of equations is encountered an enormous

e¬ort is expended to determine its symmetries. Moreover, the intuition is often expressed

that “every conservation law ought to come from some symmetry”, so whenever conserved

quantities are observed in Nature (or, more accurately, our models of Nature) people

nowadays look for a symmetry to explain it. Even when no symmetry is readily apparent,

in many cases a sort of “hidden symmetry” can be found.

Example: Motion in a Central Force Field. Consider the Lagrangian of “kinetic

minus potential energy” for an particle (of mass m = 0) moving in a “central force ¬eld”.

Here, we take Rn with its usual inner product and a function V (|x|2 ) (called the potential

energy) which depends only on distance from the origin. The Lagrangian is

2 |p| ’ V (|x|2 ).

2

m

L(x, p) =

The function EL is given by

2 |p|

2

+ V (|x|2 ),

m

EL (x, p) =

and ωL = m pi dxi = m p · dx.

The Lagrangian L is clearly symmetric with respect to rotations about the origin. For

example, the rotation in the ij-plane is generated by the vector ¬eld

‚ ‚

’ xi j .

Xij = xj

‚xi ‚x

According to Noether™s Theorem, then, the functions

µij = ωL (Xij ) = m xj pi ’ xi pj

are constant on all solutions. These are usually called the “angular momenta”. It follows

from their constancy that the bivector ξ = y(t)§y(t) is constant on any solution x = y(t) of

™

the Euler-Lagrange equations and hence that y(t) moves in a ¬xed 2-plane. Thus, we are

essentially reduced to the case n = 2. In this case, for constants E0 and µ0 , the equations

2 |p| m(x1 p2 ’ x2 p1 ) = µ0

2

+ V (|x|2 ) = E0

m

and

L.4.8 68

will generically de¬ne a surface in T R2 . The solution curves to the Euler-Lagrange equa-

tions

2

p = ’ V (|x|2 )x

x=p

™ and ™

m

which lie on this surface can then be analysed by phase portrait methods. (In fact, they

can be integrated by quadrature.)

Example: Riemannian metrics with Symmetries. As another example, consider

the case of a Riemannian manifold with in¬nitesimal symmetries. If the ¬‚ow of X on M

preserves a Riemannian metric L, then, in local coordinates,

L = gij (x)pi pj

and

‚

X = ai (x)

.

‚xi

According to Conservation of Energy and Noether™s Theorem, the functions

EL = gij (x)pi pj ωL (X ) = 2gij (x)ai (x)pj

and

are ¬rst integrals of the geodesic equations.

For example, if a surface S ‚ R3 is a surface of revolution, then the induced metric

can locally be written in the form

I = E(r) dr2 + 2 F (r)dr dθ + G(r) dθ2

where the rotational symmetry is generated by the vector ¬eld X = ‚/‚θ. The following

functions are then constant on solutions of the geodesic equations:

™™ ™ ™

E(r) r2 + 2 F (r)r θ + G(r) θ2

™ and F (r)r + G(r) θ.

™

This makes it possible to integrate by quadratures the geodesic equations on a surface of

revolution, a classical accomplishment. (See the Exercises for details.)

Subexample: Left Invariant Metrics on Lie Groups. Let G be a Lie group and let

ω 1 , ω 2 , . . . , ω n be any basis for the left-invariant 1-forms on G. Consider the Lagrangian

2 2

+ · · · + (ω n ) ,

L = ω1

which de¬nes a left-invariant metric on G. Since left translations are symmetries of this

metric and since the ¬‚ows of the right-invariant vector ¬elds Yi leave the left-invariant

1-forms ¬xed, we see that these generate symmetries of the Lagrangian L. In particular,

the functions EL = L and

µi = ω 1 (Yi )ω 1 + · · · + ω n (Yi )ω n

L.4.9 69

are functions on T G which are constant on all of the geodesics of G with the metric L. I

will return to this example several times in future lectures.

Subsubexample: The Motion of Rigid Bodies. A special case of the Lie group example

is particularly noteworthy, namely the theory of the rigid body.

A rigid body (in Rn ) is a (¬nite) set of points x1 , . . . , xN with masses m1 , . . . , mN

such that the distances dij = |xi ’ xj | are ¬xed (hence the name “rigid”). The free motion

of such a body is governed by the “kinetic energy” Lagrangian

m1 mN

|p1 |2 + · · · + |pN |2 .

L=

2 2

where pi represents the velocity of the i™th point mass. Here is how this can be converted

into a left-invariant Lagrangian variational problem on a Lie group:

Let G be the matrix Lie group

Ab

A ∈ O(n), b ∈ Rn

G= .

01

Then G acts as the space of isometries of Rn with its usual metric and thus also acts on

the N -fold product

YN = Rn — Rn — · · · — Rn

by the “diagonal” action. It is not di¬cult to show that G acts transitively on the simul-

taneous level sets of the functions fij (x) = |xi ’ xj |. Thus, for each symmetric matrix

∆ = (dij ), the set

M∆ = {x ∈ YN | |xi ’ xj | = dij }

is an orbit of G (and hence a smooth manifold) when it is not empty. The set M∆ is said to

be the “con¬guration space” of the rigid body. (Question: Can you determine a necessary

and su¬cient condition on the matrix ∆ so that M∆ is not empty? In other words, which

rigid bodies are possible?)

Let us suppose that M∆ is not empty and let x ∈ M∆ be a “reference con¬guration”

¯

which, for convenience, we shall suppose has its center of mass at the origin:

¯

mk xk = 0.

(This can always be arranged by a simultaneous translation of all of the point masses.)

Now let γ: [a, b] ’ M∆ be a curve in the con¬guration space. (Such curves are often called

“trajectories”.) Since M∆ is a G-orbit, there is a curve g: [a, b] ’ G so that γ(t) = g(t) · x.

¯

Let us write

γ(t) = (x1 (t), . . . , xN (t))

and let

A(t) b(t)

g(t) = .

0 1

L.4.10 70

The value of the canonical left invariant form on g is

A’1 A A’1 b

™ ™

± β

’1

g g=

™ = .

0 0 0 0

The kinetic energy along the trajectory γ is then

™x ™ ™x ™

mk |xk |2 = mk (xk · xk ) = mk A¯ k + b · A¯ k + b .

1 1 1

™ ™ ™