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 non-separable Hamiltonians, I show that the precession error in
symplectic integrators can be computed analytically. It is found that at each
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 exactly cancel after each period. Thus
the physics of symplectic integrators determines the optimal algorithm for
integrating long time periodic motions.Comment: 18 pages, 5 figures, 1 tabl