Explicit, time-reversible and symplectic integrator for Hamiltonians in isotropic uniformly curved geometries

TL;DR

This paper develops an explicit, time-reversible, and symplectic integrator tailored for Hamiltonian systems on curved geometries like the sphere, overcoming non-separability issues of the kinetic term.

Contribution

The paper introduces a novel explicit symplectic integrator for Hamiltonians on curved surfaces, utilizing the kinetic term's hierarchy to achieve higher-order accuracy.

Findings

The second-order scheme is explicit, time-reversible, and symplectic.

Iterative application yields a fourth-order integrator.

The method demonstrates efficiency in numerical experiments.

Abstract

The kinetic term of the $N$-body Hamiltonian system defined on the surface of the sphere is non-separable. As a result, standard explicit symplectic integrators are inapplicable. We exploit an underlying hierarchy in the structure of the kinetic term to construct an explicit time-reversible symplectic scheme of second order. We use iterative applications of the method to construct a fourth order scheme and demonstrate its efficiency.