Dissipative Perturbations of 3d Hamiltonian Systems

TL;DR

This paper explores dissipative modifications of 3D Hamiltonian systems using a metriplectic framework, ensuring energy conservation while dissipating a Casimir function, with applications to systems like a relaxing rigid body.

Contribution

It introduces a method to construct a symmetric tensor g for dissipative perturbations that preserve Hamiltonian energy and dissipate Casimir functions in 3D systems.

Findings

Construction of a symmetric tensor g satisfying specific conditions

Preservation of Hamiltonian function H during dissipation

Application to a relaxing rigid body example

Abstract

In this article we present some results concerning natural dissipative perturbations of 3d Hamiltonian systems. Given a Hamiltonian system dx/dt = PdH, and a Casimir function S, we construct a symmetric covariant tensor g, so that the modified (so-called 'metriplectic') system dx/dt = PdH + gdS satisfies the following conditions: dH is a null vector for g, and dS(gdS)< 0. Along solutions to a dynamical system of this type, the Hamiltonian function H is preserved while the function S decreases, i.e. S is dissipated by the system. We are motivated by the example of a relaxing rigid body by P.J. Morrison in which systems of this type were introduced.