Stochastic Variational Integrators

TL;DR

This paper develops stochastic variational integrators for Hamiltonian systems on manifolds, deriving stochastic equations from a variational principle, and demonstrating their symplectic and momentum-preserving properties with applications to rigid-body dynamics.

Contribution

It introduces stochastic variational integrators based on a stochastic variational principle, extending geometric integration methods to stochastic Hamiltonian systems on manifolds.

Findings

SVIs are almost surely symplectic.

SVIs preserve momentum maps in symmetric systems.

First-order mean-square convergence of SVIs on Lie groups.

Abstract

This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentum-map preserving. A first-order mean-square convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid-bodies interacting via a potential.