Variational integrators for stochastic Hamiltonian systems on Lie groups: properties and convergence

TL;DR

This paper develops structure-preserving variational integrators for stochastic Hamiltonian systems on Lie groups, ensuring properties like symplecticity and conservation laws, with proven convergence and numerical demonstrations on rigid body models.

Contribution

It introduces a novel variational integrator framework for stochastic Hamiltonian systems on Lie groups, including a convergence proof and applications to rigid body dynamics.

Findings

The integrator preserves symplectic and Lie-Poisson structures.

Convergence is proven for the rotation group.

Numerical simulations validate the method's effectiveness.

Abstract

We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and conservation of Casimir functions, are discussed, along with a discrete Noether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.