Symplectic methods for non-canonical Hamiltonian systems

TL;DR

This paper demonstrates that symplectic integrators designed for canonical Hamiltonian systems are effectively conjugate symplectic for non-canonical systems, enabling their use in long-term simulations with improved accuracy.

Contribution

It establishes that canonical symplectic methods can be applied to non-canonical Hamiltonian systems via conjugation, simplifying method development.

Findings

Symplectic methods outperform non-symplectic methods in accuracy.

Symplectic methods better preserve energy over long simulations.

Numerical experiments confirm theoretical advantages.

Abstract

We show that, when applied to any non-canonical Hamiltonian system, any integrator that is symplectic for canonical Hamiltonian problems is actually conjugate symplectic for the non-canonical structure. This result is useful because it implies that canonically symplectic methods may be successfully applied to long-time integrations of non-canonical Hamiltonian problems, thus avoiding the need to construct ad hoc new methods. Numerical results for three non-canonical Hamiltonian systems demonstrate that (canonically) symplectic methods have significant advantages in numerical accuracy and near energy preservation over non-symplectic methods.