Multisymplectic structure of nonintegrable Henon-Heiles system
A. V. Tsiganov
TL;DR
The paper investigates the existence and construction of a second invariant symplectic form for select Hamiltonian systems to enable multisymplectic discretizations. By solving invariance equations $\mathcal{L}_X P'=0$ across the nonintegrable Hénon–Heiles system, the superintegrable Kepler problem, and Toda lattices, it derives invariant bivectors and the corresponding symplectic forms, uncovering that HH and open Toda lattices admit nontrivial second invariants while periodic Toda does not. It develops two families of Hamiltonian flows preserving two symplectic forms, illustrates action-angle and Delaunay structures to express these invariants, and highlights both local and global aspects of the resulting bi-Hamiltonian frameworks. The work lays groundwork for numerical integrators that simultaneously preserve multiple geometric invariants, potentially improving long-term accuracy and stability for near-integrable and integrable 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.