Noise and dissipation on coadjoint orbits

TL;DR

This paper develops stochastic dissipative models on coadjoint orbits, analyzes their dynamics, and demonstrates the existence of random attractors with applications to rigid body systems.

Contribution

It introduces a framework for stochastic dissipation on coadjoint orbits, including new analysis of random attractors and applications to classical mechanical systems.

Findings

Existence of random attractors for semi-simple Lie algebras with positive Lyapunov exponent

Stochastic integrable reductions of the free rigid body and heavy top

Numerical simulations illustrating the behavior of random attractors

Abstract

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect-product extension. Random attractors are found for this general class of systems when the Lie algebra is semi- simple, provided the top Lyapunov exponent is positive. We study two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.