Metriplectic formalism: friction and much more

TL;DR

This paper explores the metriplectic formalism, combining Hamiltonian and dissipative dynamics, to control a rigid body's angular velocity and demonstrate convergence to stable states, with potential applications to complex systems.

Contribution

It presents a novel application of the metriplectic formalism to control theory, showing how non-Hamiltonian effects can induce convergence to stable states and attractors.

Findings

The formalism can drive a rigid body to align its angular velocity with a principal axis.

It demonstrates convergence to a limit cycle as a non-zero dimensional attractor.

Potential extension to chaotic systems and complex models is suggested.

Abstract

The metriplectic formalism couples Poisson brackets of the Hamiltonian description with metric brackets for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a preselected equilibrium state. Phenomena such as friction, electric resistivity, thermal conductivity and collisions in kinetic theories are well represented in this framework. In this paper we present an application of the metriplectic formalism of interest for the theory of control: a suitable torque is applied to a free rigid body, which is expressed through a metriplectic extension of its "natural" Poisson algebra. On practical grounds, the effect is to drive the body to align its angular velocity to rotation about a stable principal axis of inertia, while conserving its kinetic energy in the process. On theoretical grounds, this example shows how the non-Hamiltonian part of a metriplectic system may include convergence to a limit cycle, the first example of a non-zero dimensional attractor in this formalism. The method suggests a way to extend metriplectic dynamics to systems with general attractors, e.g. chaotic ones, with the hope of representing bio-physical, geophysical and ecological models.