Discrete variational calculus for double-bracket dissipation

TL;DR

This paper develops a discrete variational integrator for mechanical systems with double-bracket dissipation, preserving coadjoint orbits and decreasing energy, applicable to various physical systems.

Contribution

It introduces a geometric integrator that exactly preserves coadjoint orbits while modeling energy dissipation in forced Euler-Poincaré and Lie-Poisson systems.

Findings

The integrator preserves coadjoint orbits exactly.

It effectively models energy decrease in dissipative systems.

Numerical simulations show advantages over other methods.

Abstract

Discrete variational methods show excellent performance in numerical simulations of mechanical systems. In this paper, we adapt discrete variational integrators for the case of mechanical systems with double-bracket dissipation. In particular, we will work with forced Euler-Poincar\'e and forced Lie-Poisson systems, and the case of interest for us will be when the coadjoint orbits remain invariant, but the energy is decreasing along the orbit. This particular kind of dissipative system appears in various physical systems such as satellites with dampers, geophysical fluids, plasma physics and stellar dynamics. The proposed geometric integrator preserves the coadjoint orbits exactly. We highlight the advantages of this feature by comparing it with other general-purpose methods (including higher-order ones) across different numerical simulations.