LECTURE 5. VARIATIONAL INTEGRATORS 409So there are two generally distinct 1-forms, but (up to sign) only one 2-form. If we
make the definitionnL =de£= -<let,
then (5.32) becomes <l_)*[lL = nL. Equation (5.29), in coordinates, gives" HL -- --. 82JL -. dqO i /\ dql, j
8q(i8qi
which agrees with the discrete symplectic form discussed earlier.Noether's Theorem. Suppose a Lie group G with Lie algebra g acts on Q, and
hence diagonally on Q x Q, and that lL is G-invariant. Clearly, §is also G-invariant
and G sends critical points of § to themselves. Thus, the action of G restricts to
the space of solutions, the map <Pis G-equivariant, and from (5.31),0 = ~QxQ _J d § = ~QxQ _Jez+ foxQ _J (<P*et),
for all~ E g , or equivalently, using equivariance of <P,
~QxQ _Jez = -<P*(foxQ _J e+). (5.34)Since lL is G-invariant, (5.33) gives ~QxQ _Jez = -foxQ _Jet, which in turn
converts (5.34) to the conservation equation
(5.35)Defining the discrete momentum to be
lt=. = ~QxQ _Jet,
we see that (5.35) becomes conservation of momentum, recovering the conservation
property of these integrators we found earlier. Marsden, Patrick and Shkoller [1998]
develop a similar program for pde's in the context of multisymplectic geometry.
Following this work, we take some first steps in this direction.
Multisymplectic geometry. We recall some aspects of multisymplectic geometry,
following Gotay, Isenberg and Marsden [1997] and Marsden and Shkoller [1997].
We let 7rxy : Y _, X be a fiber bundle over an oriented manifold X. Denote the
first jet bundle over Y by J1 (Y) and identify it with the affine bundle over Y whose
fiber over y E Y-c := 7rx~(x) consists of those linear mappings / : T xX _, TyY
satisfying
T7rxy o / =Identity on TxX.
Let dimX = n + 1 and the fiber dimension of Y be N. Coordinates on X aredenoted xμ, μ = 1, 2, ... , n, 0, and fiber coordinates on Y are denoted by yA, A =
1, ... , N. These induce coordinates vA μ on the fibers of J1 (Y). If ¢ : X _, Y is a
section of 7rxy, its tangent map at x E X , denoted T x¢, is an element of J1 (Y).p(x).
Thus, the map x f--7 T.-c<P defines a section of J1 (Y) regarded as a bundle over X.
This section is denoted j^1 (¢) and is called the first jet of¢. In coordinates, j^1 (¢)
is given by
(5.36)