LECTURE 5. VARIATIONAL INTEGRATORS 401The fiber derivative is analogous to the standard Legendre transform. The coordi-
nate expression for w is:
82JL ( ) i j
W= !'.:\ i!'.l j qk+I ,qk dqk/\dqk+l"
uqkuqk+l
Theorem 5.2. The algorithm exactly preserves the symplectic form w.
One proof of this is to simply verify it with a straightforward calculation-see
Wendlandt and Marsden [1997] for the details. Another is to derive this information
directly from the variational structure. We will come back to this shortly.
The algorithm preserves momentum. Recall that Noether's theorem states
that a continuous symmetry of the Lagrangian leads to conserved quantities, as
with linear and angular momentum. A nice way to derive these conservation laws
(the way Noether did it) is to use t he invariance of the variational principle.
Let t he discrete Lagrangian b e invariant under the action of a Lie group G on
Q, and let~ E g, the Lie algebra of G. By analogy with the continuous case, define
the discrete momentum map, .Jf : Q x Q __, g* by
(.Jf(qk+I,qk),0 := (D 2 1L(qk+1,qk),~Q(qk)) ·
Theorem 5.3. The algorithm exactly preserves the momentum map.
The discrete momentum map .Jf is equivariant with respect to t he action of G
on Q x Q and the coadjoint action of G on g*. We also note that the algorithm
can b e obtained by using -JL as a generating function. Again, one can check this
by a calculation or by invoking invariance of the variational principle-which we
will indicate below.
Construction of mechanical integrators. We show how to construct integra-
tors in a practical manner by enforcing the constraints through Lagrange multipliers.
(Wendlandt and Marsden [1 997 ] also discuss the J acobia n used to solve t he nonlin-
ear equations; the constrained coordinate formulation has a special structure that
can be exploited to increase simulation efficiency, as well as local t runcation error
and solvability.)
Assume that we have a mech anical system with a constraint manifold, QC V ,
where V is a real finite dimensional vector space, and that we have an unconstrained
Lagrangian, L : TV __, IR which , by restriction of L to TQ, defines a constrained
Lagrangian, L e : TQ ,IR. Roughly speaking, V is a containing vector space in
which the computer arithmetic will take place. In particular, coordinate charts on Q
are not chosen for this purpose. In fact, apart from the use of the containing vector
space V , the algorithms developed here are independent of t he use of coordinates
on Q.
We also assume that we h ave a vector valued constraint function, g : V ,
IRk, such that our constraint manifold is given by g-^1 (0) = Q C V , wit h 0 a regular
value of g. The dimension of V is denoted n , and t herefore, the dimension of Q is
m = n-k.
Define a discrete, unconstrained Lagrangian, lL : V x V __, IR by
JL(y, x ) = L ( y; x' y ~ x ) '
where h E IR+ is the t ime step. (This is not t he only possible choice but it is
one that leads to a second order accurate algorithm.) The unconstrained action