Stochastic Geometric Mechanics and Symmetry

Abstract

We explore the interaction between geometry, symmetry, and randomness in stochastically perturbed mechanical systems, in Hamiltonian as well as Lagrangian (variational) formulations.

On the Hamiltonian side, we discuss a stochastic Kepler problem with a noisy angular momentum vector. We show that the radial distance and speed evolve deterministically. This allows us to regularize collisional singularities in a procedure similar to Moser's regularization. We generalize this to stochastic collective Hamiltonian systems. We show that the solutions to these systems are given by the action of a Lie group valued semimartingale on the deterministic solution and consequently, along directions transverse to the group orbits, these systems retain the same dynamics as their deterministic counterpart. Furthermore, we show that these systems can be described by coupling a deterministic Hamiltonian system to a stochastic one. We also identify conditions under which the momentum map of a stochastic Hamiltonian system evolves as a martingale on a reductive coadjoint orbit.

On the variational side, we construct fixed endpoint, local, and adapted variations of semimartingales in manifolds. These variations are used to prove a stochastic version of the Fundamental Lemma of the Calculus of Variations, which is subsequently applied to studying the stochastic Hamilton-Pontryagin principle. We also provide a stochastic analogue of Noether's theorem. We treat the corresponding global form of the stochastic Hamilton-Pontryagin principle via a novel approach to global variational principles by using Stratonovich operators. We describe the reduction by symmetry of the stochastic Hamilton-Pontryagin principle. Moreover, we also discuss stochastic collective dynamics from the variational point of view. Similar to the Hamiltonian case, we show that the critical point of the stochastic action is given by the action of a Lie group valued semimartingale on the critical point of the deterministic action. We also provide a description of coupling to a Lie group on the variational side.

Additionally, we extend the theory of reduction of Stratonovich differential equations, which arises in stochastic Hamiltonian systems, to stochastic differential equations given by Schwartz operators.