Lecture 5. Variational Integrators
For conservative mechanical systems with symmetry, it is of interest to develop
numerical schemes that preserve this symmetry, so that the associated conserved
quantities are preserved exactly by the integration process. One would also like
the algorithm to preserve either the Hamiltonian or the symplectic structure-one
cannot expect to do both in general, as we shall show below. There is numerical
evidence^1 that these mechanical integrators perform especially well for long time in-
tegrations, in which chaotic dynamics can be expected. Some standard algorithms
can introduce spurious effects (such as nonexistent chaos or incorrect dissipation)
in long integration runs.^2 We use the term mechanical integrator for an algo-
rithm that respects one or more of the fundamental properties of being symplectic,
preserving energy, or preserving the momentum map.
Symplectic integrators have been successfully used in a wide range of interesting
applications from molecular dynamics to integrations of the solar system.
Basic definitions and examples. By an algorithm on a phase space P we mean
a collection of maps F 7 : P ---> P (usually depending smoothly on T E IR for smallT > 0 and z E P). Sometimes we write zk+^1 = F 7 (zk ) for the algorithm and we
write 6.t or h for the step size T. We say that the algorithm is consistent or is
first order accurate with a vector field X on P ifd~FT(z)IT=o = X(z). (5.1)
Higher order accuracy is defined similarly by matching higher order derivatives. We
say that the algorithm is convergent when
(5.2)where 'Pt is the flow of X. There are some general theorems guaranteeing conver-
gence, with an important hypothesis being stability; i. e., (Ft;nr(z ) must remain
close to z for small t and all n = 1, 2, .... We refer to Chorin, Hughes, Marsden
(^1) See, for example Channell and Scovel [1990], Marsden, O 'Reilly, Wicklin and Zombro [1991],
Pullin and Saffman [1991] and articles in Marsden, Patrick and Shadwick [1996].
(^2) See , for example, Reinha ll, Caughey, and Storti [1989].
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