408 J .E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
qi: Q x Q--+ Q x Q by F(q 1 ,qo) = (q2,q 1 ), where q2 is found from the discrete
Euler-Lagrange equations (DEL equations):
{)JL {)JL-zl(q1,qo) + -zl(q2,q1) = 0. (5.27)
uq1 uqo
In this section we work out the basic differential-geometric objects of this discrete
mechanics directly from the variational point of view.
The Lagrange 1-form. We begin by calculating d§ for variations that do not fix
the endpoints:
(5.28)
It is the last two terms that arise from the boundary variations (i.e. these are the
ones that are zero if the boundary is fixed), and so these are the terms amongst
which we expect to find the discrete analogue of the Lagrange 1-form. In fact, the
boundary terms gives the two 1-forms on Q x Q
{)JL
e.r: ( q1, qo). ( 6q1, 6qo) = -zl ( q1, qo)6qo, (5.29)
uqo
and
(5.30)and we regard the pair (e-, e +) as being the analogue of the 1-form in this situa-
tion.Symplecticity of the flow. We parametrize the solutions of the variational prin-
ciple by the initial conditions ( q 1 , q 0 ) , and restrict § to that solution space. Then
equation (5.28) becomes
(5.31)We should be able to obtain the symplecticity of qi by determining what the equa-
tion dd§ = 0 means for the right-hand-side of (5.31). At first, this does not appear
to work, since dd§ = 0 gives
(5.32)
which apparently says that qi pulls a certain 2-form back to a different 2-form. The
situation is aided by the observation that, from (5.29) and (5.30),
e.r: + et = dlL, (5.33)
and consequently,