LECTURE 5. VARIATIONAL INTEGRATORS 399Proposition 5.1. (Ge and Marsden [1988]) If an algorithm for a given Hamilton-
ian system XH with a symmetry group G is energy preserving, symplectic, momen-tum preserving and G-equivariant and if the dynamics of XH is nonintegrable on
the reduced space (in the sense spelled out in the proof) then the algorithm already
gives the exact solution of the given problem up to a time reparametrization.Proof. Suppose F t:,t is our symplectic algorithm of the type discussed above, and
consider the application of the algorithm to the reduced phase space. We assume
that the Hamiltonian H is the only integral of motion of the reduced dynamics
(i.e., all other integrals of the system have been found and taken out in the re-
duction process in the sense that any other conserved quantity (in a suitable class)
is functionally dependent on H. Since Fc,t is symplectic it is the ~t-time map of
some time-dependent Hamiltonian function K. Now assume that the symplectic
map Fc,t also conserves H for all values of ~t. Thus {H,K} = 0 = {K,H}. The
latter equation implies that K is functionally dependent on H since the flow of H
(the "true dynamics" ) had no other integrals of motion. The functional dependence
of K on H in turn implies that their Hamiltonian vector fields are parallel, so the
flow of K (the approximate solution) and the flow of H (the exact solution) must
lie along identical curves in the reduced phases space; thus the flows are equivalent
up to time reparametrization. •
Colloqually speaking, this result means that it is unlikely one can find an algo-
rithm that simultaneously conserves the symplectic structure, the momentum map,and the Hamiltonian. It is tempting (but probably wrong) to guess from this that
one can monitor accuracy by keeping track of all three. However, for small time
steps, symplectic integrators have surrogate Hamiltonians as shown by Neist-
dadt [1984]; one gets nearby exactly conserved energy functions with errors that
are exponentially small (presumably below round off errors) in the time step.
Non-symplectic algorithms that conserve both momentum and energy have
been studied by Simo and Tarnow [1992], Simo and Wong [1989] and Austin, Kr-
ishnaprasad and Wang [1993]. Dissipative effects can often be dealt with by means
of product formulas. See Armero and Simo [1992, 1993, 1996] for example.
Variational methods. Symplectic-momentum integrators can be simply and nat-
urally constructed by means of discretizations of Hamilton's principle following
ideas of Veselov [1988, 1991]. We shall explain this procedure following the exposi-
tion of Wendlandt and Marsden [1997].
The emphasis here is on simplicity: the theory and practice are easy! For
example, they can be much simpler to use and implement than generating function
methods (although the two techniques are closely related theoretically).
Variational integrators include the popular Verlet methods and shake algorithms
as special cases. The methods also handle constraints in a simple way-this is one
of the sterling features of the variational technique.
The methods generalize to pde's using multisymplectic geometry with the result
being a class of multisymplectic momentum integrators. One obtains, in a
natural way, spacetime integrators. See Marsden, Patrick and Shkoller [1998] for
details and numerical examples.
There is still lots to do in this area! A goal is to continue developing the theory
and to implement multisymplectic integrators for various interesting pde systems,
such as those for nonlinear optics (the NLS equation), MHD, ocean dynamics, etc.