1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

400 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

The discrete variational principle. We now discuss discrete variational prin-

ciples (DVP) which lead to evolution algorithms analogous to the Euler-Lagrange

equations from mechanics. As we shall see, an algorithm is implicitly determined

by the associated discrete Euler-Lagrange(DEL) equations. Let us now explain how

this works.

Given a configuration space Q , a discrete Lagrangian is a map

IL:QxQ-tR

In practice, IL is obtained by approximating a given Lagrangian, but regard IL as

given for the moment. The time step information will be contained in IL.

For a positive integer N , the action sum is the map§ : QN+I -t IR defined

by

N-1

§ = LIL (qk+l> Qk),

k=O

where Qk E Q and k is a nonnegative integer. The action sum is the discrete analog

of the action integral in mechanics.

The discrete variational principle states that the evolut ion equations ex-

tremize the action sum given fixed end points, q 0 and QN. Extremizing § over

Q1, · · · , QN-1 leads to the DEL equations:

D 2IL (qk+l, Qk) + D1IL (qk, Qk-1) = 0

for all k = 1, · · · , N -1. We can write this equation in terms of a discrete algorithm

if!:QXQ-tQX Q

defined implicitly by

i .e.,

if! (qk, Qk-1) = (Qk+ l > Qk).

If, for each q E Q, D 2!L(q, q) : TqQ -t r;Q is invertible, then D 2 IL : Q x Q -t

T *Q is locally invertible and so the algorithm if!, which flows the system forward

in discrete time, is well defined for small time steps.

In coordinates, qi on Q, the DEL equations are

8IL 8IL

~ 0 if! (Qk+l, Qk) + ~ (Qk+l, Qk) = 0

uqk uqk+l

i. e.,

The algorithm is symplectic. To explain the sense in which the algorithm is
symplectic, first define the fiber derivative by

FIL: Q x Q -t T *Q; (q1, qo) 1---t (qo, D2IL (q1, Qo))

and define the 2-form won Q x Q by pulling back the canonical 2-form on T*Q:

w =FIL* (DcAN).