Stochastic Implicit Lagrange-Poincaré Reduction

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 ${\tilde{\mathfrak{g}}}^{\ast }$. This provides a stochastic analogue of the deterministic implicit Lagrange-Poincaré reduction. The stochastic Euler-Lagrange equations drop to a set of stochastic horizontal and vertical Lagrange-Poincaré equations on $T(Q/G)\oplus T^{\ast }(Q/G)\oplus \tilde{\mathfrak{g}}\oplus {\tilde{\mathfrak{g}}}^{\ast }$. 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.