Metriplectic dynamical systems on contact manifolds

TL;DR

This paper introduces a new metriplectic dynamical system on contact manifolds that models thermodynamic behavior, exemplified by deriving the Duffing equation as a subsystem with thermodynamic properties.

Contribution

It constructs a natural metriplectic system on the jet bundle combining Poisson and contact structures, ensuring thermodynamic consistency unlike standard contact Hamiltonian systems.

Findings

The system preserves the Hamiltonian and increases entropy over time.

The Duffing equation is derived as a subsystem within a thermodynamic framework.

The approach facilitates stability analysis of equilibrium states.

Abstract

Flows on symplectic, Poisson, contact, and metriplectic manifolds are reviewed in order to describe our main result, which is to associate a natural metriplectic dynamical system on the general one-jet bundle $J^1N=T^*N\times \mathbb{R}$, which is at once a (trivial) Poisson manifold and a contact manifold. Unlike the standard contact Hamiltonian system, our metriplectic system is thermodynamically consistent in that $$\dot{H} = 0 \quad\mathrm{and}\quad \dot{S} \geq 0$$ under the flow. Here $H$ is the Hamiltonian, while $S$ is the entropy function which is nothing but the $\mathbb{R}$ coordinate function of $J^1N$. As an example we derive the Duffing equation (autonomous and nonautonomous versions) either as a contact Hamiltonian system or as a metriplectic system. We show that for both systems the Duffing equation is a subsystem of three dimensional systems that contain a thermodynamic component, a form that facilitates asymptotic stability analysis of the relevant equilibrium state.