and “(b, s) = γ(b) for all s ∈ (’µ, µ).

In this lecture, “variation” will always mean “smooth variation with ¬xed endpoints”.

If L is a Lagrangian on M and “ is a variation of γ: [a, b] ’ M, then we can de¬ne a

function FL,“ : (’µ, µ) ’ R by setting

FL,“ (s) = FL (γs )

where γs (t) = “(t, s).

De¬nition 2: A curve γ: [a, b] ’ M is L-critical if FL,“ (0) = 0 for all variations of γ.

It is clear from calculus that a curve which minimizes FL among all curves with the

same endpoints will have to be L-critical, so the search for minimizers usually begins with

the search for the critical curves.

Canonical Coordinates. I want to examine what the problem of ¬nding L-critical

curves “looks like” in local coordinates. If U ‚ M is an open set on which there exists a

coordinate chart x: U ’ Rn , then there is a canonical extension of these coordinates to a

coordinate chart (x, p): T U ’ Rn — Rn with the property that, for any curve γ: [a, b] ’ U,

with coordinates y = x —¦ γ, the p-coordinates of the curve γ: [a, b] ’ T U are given by

™

p —¦ γ = y. We shall call the coordinates (x, p) on T U , the canonical coordinates associated

™ ™

to the coordinate system x on U.

The Euler-Lagrange Equations. In a canonical coordinate system (x, p) on T U

where U is an open set in M, the function L can be expressed as a function L(x, p) of x

and p. For a curve γ: [a, b] ’ M which happens to lie in U, the functional FL becomes

simply

b

FL (γ) = L y(t), y(t) dt.

™

a

I will now derive the classical conditions for such a γ to be L-critical: Let h: [a, b] ’ Rn

be any smooth map which satis¬es h(a) = h(b) = 0. Then, for su¬ciently small µ, there is

a variation “ of γ which is expressed in (x, p)-coordinates as

™

(x, p) —¦ “ = (y + sh, y + sh).

™

L.4.2 62

Then, by the classic integration-by-parts method,

b

d ™

FL,“ (0) = L y(t) + sh(t), y(t) + sh(t) dt

™

ds s=0 a

b

‚L ‚L ™

y(t), y(t) hk (t) + k y(t), y(t) hk (t)

= ™ ™ dt

k

‚x ‚p

a

b

‚L d ‚L

y(t), y(t) ’ hk (t) dt.

= ™ y(t), y(t)

™

‚xk ‚pk

dt

a

This formula is valid for any h: [a, b] ’ Rn which vanishes at the endpoints. It follows

without di¬culty that the curve γ is L-critical if and only if y = x —¦ γ satis¬es the n

di¬erential equations

‚L d ‚L

y(t), y(t) ’ for 1 ¤ k ¤ n.

™ y(t), y(t)

™ = 0,

‚xk ‚pk

dt

These are the famous Euler-Lagrange equations.

The main drawback of the Euler-Lagrange equations in this form is that they only

give necessary and su¬cient conditions for a curve to be L-critical if it lies in a coordinate

neighborhood U. It is not hard to show that if γ: [a, b] ’ M is L-critical, then its restriction

to any subinterval [a , b ] ‚ [a, b] is also L-critical. In particular, a necessary condition for

γ to be L-critical is that it satisfy the Euler-Lagrange equations on any subcurve which lies

in a coordinate system. However, it is not clear that these “local conditions” are su¬cient.

Another drawback is that, as derived, the equations depend on the choice of coordi-

nates and it is not clear that one™s success in solving them might not depend on a clever

choice of coordinates.

In what follows, we want to remedy these defects. First, though, here are a couple of

examples.

Example: Riemannian Metrics. Consider a Riemannian metric L: T M ’ R. Then,

in local canonical coordinates,

L(x, p) = gij (x)pi pj .

where g(x) is a positive de¬nite symmetric matrix of functions. (Remember, the summa-

tion convention is in force.) In this case, the Euler-Lagrange equations are

‚gij d ‚gkj

y(t) y i (t)y j (t) = 2gkj y(t) y j (t) = 2 i y(t) y i (t)y j (t) + 2gkj y(t) y j (t).

™ ™ ™ ™ ™ ¨

‚xk dt ‚x

Since the matrix g(x) is invertible for all x, these equations can be put in more familiar

form by solving for the second derivatives to get

y i = ’“i (y)y j y k

¨ ™™

jk

L.4.3 63

where the functions “i = “i are given by the formula so familiar to geometers:

jk kj

1i ‚g j ‚g k ‚gjk

’

“i = g +

jk

‚xk ‚xj

2 ‚x

where the matrix g ij is the inverse of the matrix gij .

Example: One-Forms. Another interesting case is when L is linear on each tangent

space, i.e., L = ω where ω is a smooth 1-form on M. In local canonical coordinates,

L = ai (x) pi

for some functions ai and the Euler-Lagrange equations become:

‚ai d ‚ak

y(t) y i (t) = y(t) y i (t)

™ ak y(t) = ™

‚xk ‚xi

dt

or, simply,

‚ai ‚ak

(y) ’ (y) y i = 0.

™

‚xk ‚xi

This last equation should look familiar. Recall that the exterior derivative of ω has the

coordinate expression

1 ‚aj ‚ai

’ j dxi § dxj .

dω =

2 ‚xi ‚x

If γ: [a, b] ’ U is Fω -critical, then for every vector ¬eld v along γ the Euler-Lagrange

equations imply that

1 ‚aj ‚ai

y(t) ’ j y(t) y i (t)v j (t) = 0.

dω γ(t), v(t) =

™ ™

‚xi

2 ‚x

In other words, γ(t) dω = 0. Conversely, if this identity holds, then γ is clearly ω-critical.

™

This leads to the following global result:

Proposition 1: A curve γ: [a, b] ’ M is ω-critical for a 1-form ω on M if and only if it

satis¬es the ¬rst order di¬erential equation

γ(t) dω = 0.

™

Proof: A straightforward integration-by-parts on M yields the coordinate-free formula

b

Fω,“ (0) = ‚“

dω γ(t),

™ ‚s (t, 0) dt

a

where “ is any variation of γ and ‚“ is the “variation vector ¬eld” along γ. Since this

‚s

vector ¬eld is arbitrary except for being required to vanish at the endpoints, we see that

“dω γ, v = 0 for all vector ¬elds v along γ” is the desired condition for ω-criticality.

™

L.4.4 64

The way is now paved for what will seem like a trivial observation, but, in fact, turns

out to be of fundamental importance: It is the “seed” of Noether™s Theorem.

Proposition 2: Suppose that ω is a 1-form on M and that X is a vector ¬eld on M

whose (local) ¬‚ow leaves ω invariant. Then the function ω(X) is constant on all ω-critical

curves.

Proof: The condition that the ¬‚ow of X leave ω invariant is just that LX (ω) = 0.

However, by the Cartan formula,

0 = LX (ω) = d(X ω) + X dω,

so for any curve γ in M, we have

dω γ(t), X(γ(t)) = ’dω X(γ(t)), γ(t) = ’(X dω) γ(t) = d(X ω) γ(t)

™ ™ ™ ™

and this last expression is clearly the derivative of the function X ω = ω(X) along γ.

Now apply Proposition 1.

It is worth pausing a moment to think about what Proposition 2 means. The condition

that the ¬‚ow of X leave ω invariant is essentially saying that the ¬‚ow of X is a “symmetry”

of ω and hence of the functional Fω . What Proposition 2 says is that a certain kind of

symmetry of the functional gives rise to a “¬rst integral” (sometimes called “conservation

law”) of the equation for ω-critical curves. If the function ω(X) is not a constant function

on M, then saying that the ω-critical curves lie in its level sets is useful information about

these critical curves.

Now, this idea can be applied to the general Lagrangian with symmetries. The only

trick is to ¬nd the appropriate 1-form on which to evaluate “symmetry” vector ¬elds.

Proposition 3: For any Lagrangian L: T M ’ R, there exist a unique function EL on T M

and a unique 1-form ωL on T M which, relative to any local coordinate system x: U ’ R,

have the expressions

‚L ‚L

’L

EL = pi dxi .

and ωL =

‚pi i

‚p

Moreover, if γ: [a, b] ’ M is any curve, then γ satis¬es the Euler-Lagrange equations for

L in every local coordinate system if and only if its canonical lift γ: [a, b] ’ T M satis¬es

™

γ (t) dωL = ’dEL (γ(t)).

¨ ™

Proof: This will mainly be a sequence of applications of the Chain Rule.

There is an invariantly de¬ned vector ¬eld R on T M which is simply the radial vector

¬eld on each subspace Tm M. It is expressed in canonical coordinates as R = pi ‚/‚pi .

Now, using this vector ¬eld, the quantity EL takes the form

EL = ’L + dL(R).

L.4.5 65

Thus, it is clear that EL is well-de¬ned on T M.

Now we check the well-de¬nition of ωL . If z: U ’ R is any other local coordinate

system, then z = F (x) for some F : Rn ’ Rn . The corresponding canonical coordinates on

T U are (z, q) where q = F (x)p. In particular,

dz F (x) 0 dx

= .

dq G(x, p) F (x) dp

where G is some matrix function whose exact form is not relevant. Then writing Lz for

‚L ‚L

‚z1 , . . . , ‚zn , etc., yields

dL = Lz dz + Lq dq

= Lz F (x) + Lq G(x, p) dx + Lq F (x) dp

= Lx dx + Lp dp.

Comparing dp-coe¬cients yields Lp = Lq F (x), so Lp dx = Lq F (x) dx = Lq dz. In partic-

ular, as we wished to show, there exists a well-de¬ned 1-form ωL on T M whose coordinate

expression in local canonical coordinates (x, p) is Lp dx.

The remainder of the proof is a coordinate calculation. The reader will want to note

that I am using the expression γ to denote the velocity of the curve γ in T M. The curve

¨ ™

γ is described in U as (x, p) = (y, y) and its velocity vector γ is simply (x, p) = (y, y ).

™ ™ ¨ ™™ ™¨

Now, the Euler-Lagrange equations are just

‚2L ‚2L

‚L d ‚L

= i j (y, y)¨ + i j (y, y)y j .

j

(y, y) =

™ (y, y)

™ ™y ™™

‚xi ‚pi

dt ‚p ‚p ‚p ‚x