406 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
space CL may b e identified with the initial conditions for the fl.ow; to Vq E TQ, we
associate the integral curves 1-7 F 8 (vq), s E [O, t]. The value of 6 on that curve is
denoted by 6 t, and again called t he action. We thus regard 6 t as a real valued
function on TQ. The fundamental equat ion (5.22) b ecomes
d6t(vq)Wvq = 8L(Ft (Vq)) · : E l<=O Ft(Vq + EWvJ - 8L(vq) · Wvq'
where E 1-7 vq + EWvq symbolically represents any curve at Vq with derivative Wvq.
We have thus derived the equation
d<St = Ft8L - 8 r,. (5.23)
Taking the exterior derivative of (5.23) yields the fundamental fact that the fl.ow of
X is symplectic:
0 = dd6t = d(Ft 8 L - 8 L) = -Ft'0.L + 0.r,
which is equivalent to Ft0.L = 0.L. Thus, using the variational principle, the ana-logue that the evolution is symplectic is the equation d^2 = 0, applied to the action
restricted to space of solutions of the variational principle. Equation (5.23) also
provides the differential-geometric equations for X. Indeed , one time derivativeof (5.23) gives dL = £x8c so that
x _J n L = - X _J d8L = -£x8L + d(X _J 8 L) = d(X _J 8L - L) = dE,
if we define E := X _j 8 L - L. Thus, we quite naturally find that X =Xe.
Of course, this set up also leads directly to Ha milton-Jacobi t heory, which was
one of t he ways in which symplectic integrators were developed (see McLachlan and
Scovel [1996] and references therein.) However , we shall not pursue this aspect of
the theory.
Momentum maps. Suppose that a Lie group G, with Lie algebra g , acts on Q,
and hence on curves in Q, in such a way that the action 6 as defined by ( 5. 17)
is invariant. This is implied by, (but does not imply ) that L itself is invariant.^3
Clearly, G sends solutions of the variational principle to themselves, so the action
of G restricts to CL, and the action commutes with Ft. Denoting the infinitesimal
generator of~ Egon TQ by frQ, we have by (5.23),
0 = frQ _J d6t = frQ _J (Fi*8L - 8L) = Ft(frQ _J 8L) - frQ _J 8 L. (5.24)
For ~ E g, define Jf. : TQ __, JR by lt, = frQ _J 8 L. Then (5.24) says the lt, is
an integral of the fl.ow of X e. We have arrived at a version of Noether's theorem
(actually rather close to the original derivation of Noether): Using the variational
principle, Noether's theorem results from the infinitesimal invariance of the action
restricted to space of solutions of the variational principle. The conserved momen-
tum associated to a Lie algebra element ~ is It, = ~ _J 8 L' where 8 L is the Lagrange
one-form.
Reformulation in terms of first variations. We have seen that symplecticity
of the fl.ow and Noether's theorem result from restricting the action to the space
of solutions. The tacit assumption is that the sp ace of solutions is a ma nifold in
some appropriate sense. This is a potential problem, since solution spaces for field
theories are known to have singularities (see, eg, Arms, Marsden and Moncrief
[1982]). We now show how this problem can b e avoided.
(^3) The distinction is sometimes important. See Olver [1986] for a discussio n.