Physics of symplectic integrators: perihelion advances and symplectic corrector algorithms

Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for nonseparable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at leading order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors cancel at that order after each period. With the use of correctable algorithms, both the energy and precession error are of effective order n+2 where n is the nominal order of the algorithm. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long-time periodic motions.