Stochastic Implicit Lagrange-Poincar\'e Reduction

TL;DR

This paper develops a stochastic version of the Lagrange-Poincaré reduction for Hamilton-Pontryagin principles, providing new equations for stochastic mechanical systems with symmetry, including applications to rigid bodies and charged particles.

Contribution

It introduces a stochastic implicit Lagrange-Poincaré reduction framework, extending deterministic reduction to stochastic systems on manifolds with symmetry.

Findings

Derived stochastic horizontal and vertical Lagrange-Poincaré equations.

Applied the theory to stochastic rigid body and charged particle models.

Established a stochastic analogue of classical reduction principles.

Abstract

In this paper we consider reduction of the stochastic Hamilton-Pontryagin principle formulated on the Pontryagin bundle of a manifold $Q$. We prove that a stochastic action invariant under the free and proper action of a Lie group $G$ drops to a reduced variational principle expressed in terms of variables of the Pontryagin bundle of the reduced space $Q/G$, the associated adjoint bundle $\tilde{\mathfrak{g}}:= (Q\times \mathfrak{g})/G$ and its dual bundle $\tilde{\mathfrak{g}}^*$. This provides a stochastic analogue of the deterministic implicit Lagrange-Poincar\'e reduction. The stochastic Euler-Lagrange equations drop to a set of stochastic horizontal and vertical Lagrange-Poincar\'e equations on $T(Q/G)\oplus T^*(Q/G)\oplus\tilde{\mathfrak{g}}\oplus\tilde{\mathfrak{g}}^*$. As examples, we consider stochastic perturbations of the rigid body with a rotor, as well as a Kaluza-Klein description of stochastic perturbations of a charged particle in a magnetic field.