Metriplectic Heavy Top: An Example of Geometrical Dissipation

TL;DR

This paper explores the geometric formulation of dissipative dynamics using a metriplectic framework, specifically applying it to a heavy top with Euclidean curvature, revealing natural relaxation to principal axis rotation.

Contribution

It introduces a geometric approach to dissipative systems via metriplectic brackets and applies it to a heavy top with curvature, generalizing previous energy-dissipation models.

Findings

The equations cause the top to relax to a principal axis rotation.

The geometric brackets naturally generalize energy-conserving dissipation.

Dissipative dynamics are described using curvature-inspired brackets.

Abstract

Recently, Morrison and Updike showed that many dissipative systems are naturally described as possessing a Riemann curvature-like bracket, which similar to the Poisson bracket, generates the dissipative equations of motion once suitable generators are chosen. In this paper, we use geometry to construct and explore the dynamics of these new brackets. Specifically, we consider the dynamics of a heavy top with dissipation imposed by a Euclidian contravariant curvature. We find that the equations of motion, despite their rather formal motivation, naturally generalize the energy-conserving dissipation considered by Matterasi and Morrison. In particular, with suitable initial conditions, we find that the geometrically motivated equations of motion cause the top to relax to rotation about a principal axis.