Metriplectic Euler-Poincaré equations: smooth and discrete dynamics

TL;DR

This work develops a metriplectic extension of Euler-Poincaré and Lie-Poisson dynamics by incorporating a dissipative symmetric bracket that preserves the total energy $H$ while enforcing entropy production through a Casimir-like function $S$. It establishes a geometric framework with a Poisson structure and a positive semidefinite inner product ${\mathcal K}$, analyzes symmetry via momentum maps, and derives forced Euler-Poincaré equations that relax towards increased entropy. It then designs second-order, discrete-gradient based integrators that exactly preserve energy and guarantee nonnegative entropy production, extending the method to differentiable manifolds using retractions. A detailed example on the relaxing rigid body demonstrates both the continuous metriplectic dynamics and the structure-preserving numerics, highlighting practical viability for dissipative geometric mechanics.

Abstract

In this paper we will study some interesting properties of modifications of the Euler-Poincaré equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism. The metriplectic representation of the dynamics allows us to describe the conservation of energy, as well as to guarantee entropy production. Moreover, we describe the use of discrete gradient systems to numerically simulate the evolution of the continuous metriplectic equations preserving their main properties: preservation of energy and correct entropy production rate.

Figures (1)

• Figure 1: $I_1=10, I_2=5, I_3=1, h=0.1$, initial conditions $p_1=0.001, p_2=-1, p_3=0.001$

Theorems & Definitions (8)

• Remark 1
• Example 1
• Example 2
• Definition 1
• Proposition 2
• proof
• Proposition 3
• proof