Multisymplectic structure of nonintegrable Henon-Heiles system

TL;DR

This paper introduces a second invariant symplectic form for the nonintegrable Henon-Heiles system and other benchmark problems, enabling the construction of multi-symplectic integrators that preserve multiple symplectic structures.

Contribution

It presents a novel second invariant symplectic form for nonintegrable systems, facilitating multi-symplectic integrator development beyond traditional single-form approaches.

Findings

Successfully constructs multi-symplectic integrators for benchmark systems.

Demonstrates preservation of multiple symplectic forms in numerical simulations.

Extends multi-symplectic methods to nonintegrable Hamiltonian systems.

Abstract

Multi-symplectic integrators are typically regarded as a discretization of the Hamiltonian partial differential equations. This is due to the fact that, for generic finite-dimensional Hamiltonian systems, there exists only one independent symplectic structure. In this note, the second invariant symplectic form is presented for the nonintegrable Henon-Heiles system, Kepler problem, integrable and non-integrable Toda type systems. This approach facilitates the construction of a multi-symplectic integrator, which effectively preserves both symplectic forms for these benchmark problems.