New numerical integrator for the Stäckel system conserving the same number of constants of motion as the degree of freedom

Abstract

A new numerical integrator for a wide class of separable Hamiltonian systems called the Stackel system is presented. This integrator is designed so as to conserve the same number of constants of motion as the degree of freedom of the Stackel system. Separation of variables of the Stackel system is most fundamental for the integrator. A combination of canonical transformations and an energy-preserving method with a variable step-size plays a key role to design such an integrator. Some typical and important examples of the Stackel system are then discretized explicitly. They are the three-dimensional Kepler motion, the Holt system and the integrable Henon–Heiles system in celestial mechanics.