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

François Gay-Balmaz, Meng Wu

TL;DR

The paper develops stochastic variational integrators for Hamiltonian systems on Lie groups by discretizing the stochastic Hamilton phase-space principle, yielding a stochastic midpoint scheme that is symplectic and respects Lie-Poisson reduction. It extends the vector-space midpoint method to Lie groups via a retraction map, proving symplecticity and discrete Noether properties, including momentum preservation and Casimir conservation under symmetry. For advected-parameter problems (semidirect products), reduced formulations preserve coadjoint orbits and Lie-Poisson structure, with a Noether theory adapted to the advection. A full convergence proof is established for the SO(3) case (rigid body), with numerical demonstrations on the free rigid body and heavy top, illustrating structure preservation and ensemble forecasting capabilities under stochastic forcing. The framework provides a unified approach to structure-preserving discretization of stochastic Hamiltonian systems, with potential applications to stochastic geometric fluids and related dynamics.

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.

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

TL;DR

The paper develops stochastic variational integrators for Hamiltonian systems on Lie groups by discretizing the stochastic Hamilton phase-space principle, yielding a stochastic midpoint scheme that is symplectic and respects Lie-Poisson reduction. It extends the vector-space midpoint method to Lie groups via a retraction map, proving symplecticity and discrete Noether properties, including momentum preservation and Casimir conservation under symmetry. For advected-parameter problems (semidirect products), reduced formulations preserve coadjoint orbits and Lie-Poisson structure, with a Noether theory adapted to the advection. A full convergence proof is established for the SO(3) case (rigid body), with numerical demonstrations on the free rigid body and heavy top, illustrating structure preservation and ensemble forecasting capabilities under stochastic forcing. The framework provides a unified approach to structure-preserving discretization of stochastic Hamiltonian systems, with potential applications to stochastic geometric fluids and related dynamics.

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.
Paper Structure (40 sections, 15 theorems, 201 equations, 14 figures)

This paper contains 40 sections, 15 theorems, 201 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Previous works.
  3. Content of the paper.
  4. Stochastic variational integrators on vector spaces
  5. Stochastic phase space principle on vector spaces
  6. Discrete stochastic phase space principle on vector spaces
  7. Stochastic variational integrators on Lie groups
  8. Stochastic Hamilton phase space principle on Lie groups
  9. Discrete stochastic phase space principle on Lie groups
  10. Symplecticity
  11. Symmetries, discrete momentum maps, coadjoint orbits, and Casimirs
  12. Momentum maps and discrete Noether's theorem
  13. Preservation of coadjoint orbits, Casimirs, and Lie-Poisson brackets
  14. Hamiltonians with an advected parameter and semidirect products
  15. Stochastic Lie group variational integrators with advected parameter
  16. ...and 25 more sections

Key Result

Proposition 2.2

Consider the critical condition for the discrete Hamilton phase space principle where $c_d$ is the discrete curve in the form discete_curve with the end points $q_0$ and $q_K$ fixed and $\mathcal{G} _d$ the discrete action functional defined in discrete_PS2. Denote with $\mathcal{S}_d$ the extremum value obtained from the variational principle for discrete curves with fixed $(q Define $p_0$ and $

Figures (14)

  • Figure 1: Two stochastic paths on the angular momentum sphere, with the initial condition $\Pi_0 = (-0.5878, 0, 0.8090)$, marked by the red dot. The deterministic path of the same initial condition is highlighted in red.
  • Figure 2: With the initial condition $\Pi_0 = (-0.5878, 0, 0.8090)$, an ensemble of 20 stochastic paths are generated. Each image shows the positions of $\Pi_k$ of the ensemble at the given time horizon. The deterministic $\Pi_k$ is marked in red as reference. The distances between the forecasts of the stochastic paths tend to increase as time increases. The possibility of bifurcation significantly changes the scattering pattern.
  • Figure 3: $V(\theta)$ as function of $\theta$, the dashed line marks the level of $E'$, with parameters of the deterministic case discussed below.
  • Figure 4: Deterministic
  • Figure 5: Deterministic
  • ...and 9 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 17 more