Local Universal Splitting Integrators for Contact Hamiltonian Systems

TL;DR

This paper introduces a geometric splitting framework for contact Hamiltonian systems, enabling structure-preserving integrators that are locally universal and can be numerically realized via symplectic and ODE integrators.

Contribution

It develops a novel splitting method for contact Hamiltonian systems based on exact-contact subflows, establishing local universality and practical numerical realization.

Findings

Lie algebra generated by strict and prolonged Hamiltonians is dense in all smooth contact Hamiltonians.

The framework provides contact splitting integrators constructed from exact subflows.

Numerical examples demonstrate the effectiveness of the proposed integrators.

Abstract

Contact Hamiltonian systems extend symplectic Hamiltonian mechanics to dissipative settings while retaining geometric structure. We develop a structure-preserving splitting framework for contact Hamiltonian systems on $J^1(\mathbb{R}^n)$ based on two tractable classes of exact-contact subflows: strict contactomorphisms and prolonged diffeomorphisms. Our main theoretical result is that the Lie algebra generated by the corresponding strict and prolonged Hamiltonians contains all polynomial-in-$p$ Hamiltonians and is therefore dense, in the $C^r$ topology on compact sets, in the Lie algebra of smooth contact Hamiltonians. This yields a local universality result and contact splitting integrators built from exact strict and prolonged subflows. We then show how these subflows can be realized numerically by lifting symplectic integrators on $T^*\mathbb{R}^n$ and ODE integrators on $\mathbb{R}^n\times\mathbb{R}$. Finally, we illustrate the framework on a sequence of low-dimensional examples.