Jacobi Hamiltonian Integrators: construction and applications

TL;DR

This paper introduces a systematic framework for creating geometric integrators tailored for Hamiltonian systems on Jacobi manifolds, enhancing structure preservation and long-term numerical stability.

Contribution

It develops a novel method combining Poissonization and symplectic realizations to construct explicit Jacobi Hamiltonian integrators for various systems.

Findings

Integrators better preserve geometric structures.

Improved long-term numerical stability.

Effective for contact and classical Hamiltonian models.

Abstract

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.