Symplectic integrators revisited

Abstract

This paper is a survey on Symplectic Integrator Algorithms (SIA): numerical integrators designed for Hamiltonian systems. As it is well known, n degrees of freedom Hamiltonian systems have an important property: their flows preserve not only the total volume of the phase space, which is only one of the Poincaré invariants, but also the volume of sub-spaces less then 2n. These invariants are inherited from the conservation of the symplectic area. It is usually demanded of integrators that they should preserve energy. In this survey the main point is to convince the readers that the preservation of the symplectic area or canonicity of the Hamiltonian flow can be equally important, mainly when the concern is not one particular trajectory but the behavior of the phase space as a whole for long intervals of time. The KAM theorem asserts that for any integrable Hamiltonian perturbed by a small Hamiltonian term, such as that caused by the construction of the SIA, the perturbed dynamics preserves most of the incommensurate, nondegenerate, invariant tori. Unstable objects and their invariant manifolds are structurally stable and will be well represented by symplectic integrators.