Collective Symplectic Integrators on $S_2^N \times T^*\mathbb{R}^M$

TL;DR

This paper introduces a new symplectic integrator tailored for Hamiltonian systems on product manifolds, expanding the toolkit for accurate long-term simulations in geometric mechanics.

Contribution

It develops a novel symplectic integrator for systems on $S_2^n imes T^* r^m$ and derives algebraic conditions for partitioned Runge--Kutta methods to be symplectic.

Findings

The integrator is effective for Hamiltonian equations on specified manifolds.

Algebraic conditions for symplecticity of partitioned Runge--Kutta methods are established.

The approach enhances numerical stability and accuracy for complex Hamiltonian systems.

Abstract

A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically, algebraic conditions for their symplecticity are derived.