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Stochastic Variational Integrators

Nawaf Bou-Rabee, Houman Owhadi

arXiv:0708.2187v2math.PR

TL;DR

The paper addresses the unproven converse linking critical points of stochastic actions to stochastic Hamilton’s equations. It develops stochastic variational principles and integrators for systems on manifolds; its main theorem establishes that action characterization, and a pendulum simulation reports correct temperature for the SVI where Euler-Maruyama schemes heat or cool the system.

  • Problem

    For stochastic Hamiltonian systems on manifolds, prior work had not established that critical points of a stochastic action satisfy stochastic Hamilton’s equations, and a counterexample exists for a particular action.

    Source: §1 · p. 3

  • Method

    The paper extends Hamilton-Pontryagin variational principles to stochastic systems and discretizes variational principles to construct stochastic variational integrators.

    Source: §1 · pp. 3–4

  • Results

    The paper proves an action characterization for a class of stochastic Hamiltonian systems; in a ballistic-pendulum simulation, the SVI computes correct temperature while Euler-Maruyama schemes artificially heat or cool the system.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    The paper develops a first-order mean-squared stochastic variational Euler method for mechanical systems on Lie groups.

    Source: §1 · p. 4

    The analysis restricts admissible curves to C1 and stochastic forcing to Wiener processes, although the authors say the ideas can extend to more general semimartingales.

    Source: §1 · p. 3

Abstract

from arXiv · show

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.

1 Introduction

The paper extends variational mechanics to stochastic Hamiltonian systems on manifolds, addressing a gap in the action-based characterization of their dynamics. It develops stochastic variational integrators and illustrates their behavior in long-time simulations.

1 Introduction

  • Structure-preserving integrators motivate stochastic extensions because standard methods can corrupt chaotic invariant sets, and variational integrators have been reported to avoid long-time drift in a lattice’s time-averaged temperature.The paper’s stated goal is efficient, structure-preserving integration for long-time simulations of stochastic Hamiltonian systems on manifolds.Source: §1 · pp. 1–2
  • Earlier work left unproven the converse from critical points of a stochastic action to stochastic Hamilton’s equations, and a counterexample exists for one choice of action.The paper frames its theorem as addressing this gap.Source: §1 · p. 3
  • The paper proves that, for a class of mechanical systems on paracompact manifolds with multiplicative white-noise forces and torques, a.s. solutions of stochastic Hamilton’s equations are exactly the curves extremizing a stochastic action.The paper identifies this theorem as its main result.Source: §1 · p. 3
  • The paper derives SDEs for holonomically constrained systems and systems with nonconservative drift, and performs stochastic Lagrangian reduction on Lie groups.The derivations use constrained stochastic variational and Lagrange-d’Alembert-type principles; the paper obtains stochastic Euler-Poincaré/Lie-Poisson equations.Source: §1 · p. 3
  • Discretized stochastic variational principles yield SVIs, stochastic RATTLE-type methods, and stochastic Euler-Poincaré/Lie-Poisson integrators.The paper also describes quasi-symplectic methods for rigid-body-type systems at uniform temperature.Source: §1 · p. 3
  • A single-step stochastic variational Euler method is developed for Lie-group systems, with the paper’s rigid-body application shown to be first order in the mean-squared sense.For translation-group systems, the paper relates its methods to stochastic Hamiltonian systems and, with dissipation in the drift, to Langevin equations.Source: §1 · p. 4
  • In the ballistic-pendulum simulation, the SVI computes the correct temperature, while explicit and implicit Euler-Maruyama schemes artificially heat and cool the system, respectively.All methods are first-order strongly convergent and use the same Wiener-process realization; the paper presents this as a suggestion of favorable long-time energy behavior.Source: §1 · p. 4

2 Stochastic Flows on Manifolds

This section sets out the stochastic-flow framework used for systems on manifolds. It adopts Stratonovich SDEs and recalls conditions ensuring existence, uniqueness, and mean-squared smoothness of the flow.

2 Stochastic Flows on Manifolds

  • A stochastic dynamical system combines a probability-space base flow that propagates noise with a stochastic flow on the manifold.The base flow is P-preserving, and the manifold flow is described using identity and cocycle properties.Source: §2 · p. 5
  • The paper adopts Stratonovich stochastic integrals for SDEs on manifolds because their differential obeys the chain rule.The SDEs use a drift vector field and diffusion vector fields driven by independent Wiener processes.Source: §2 · p. 5
  • Under the stated uniformly Lipschitz and measurability conditions, each initial point has an a.s. solution curve, and that solution is pathwise unique.The existence and uniqueness result is stated for vector fields of class X^k(M), k ≥ 1.Source: §2 · p. 6
  • The resulting stochastic flow is a.s. C^k in the initial condition in the mean-squared sense and C^0 in time.Mean-squared differentiability is defined using a norm on the model space and extended locally to the manifold.Source: §2 · p. 6

3 Stochastic HP Mechanics

The stochastic Hamilton–Pontryagin (HP) principle makes stochastic HP equations on manifolds equivalent almost surely to criticality of a stochastic action. The section extends this variational framework to symplecticity, symmetry, constraints, nonconservative forces, and Lie-group reduction.

Stochastic HP Principle

  • The stochastic action perturbs the classical HP action with Stratonovich integrals, and variations are taken over curves while the probability space is fixed.The HP formulation varies q, v, and p independently, with q fixed at the endpoints.Source: §3 · pp. 6–7
  • Almost surely, a curve satisfies the stochastic HP equations if and only if it is a critical point of the stochastic action.The theorem assumes a C2 Lagrangian and stochastic potentials with globally Lipschitz first derivatives.Source: §3 · p. 7

Structure Preservation

  • Provided v can be eliminated using the Legendre transform, the stochastic flow preserves the canonical symplectic form almost surely.When the action is infinitesimally symmetric, the corresponding momentum map is also conserved almost surely.Source: §3 · pp. 11–12

Holonomic Constraints

  • For a stochastic Hamiltonian system with invertible ∂2L/∂v2 and holonomic constraint g:Q→Rk, Theorem 3.6 gives an equivalence involving constrained stochastic HP equations; the flow of (3.3) is mean-square symplectic.The constraint is g:Q→Rk with S=g−1(0).Source: §3 · p. 13

Nonconservative Effects

  • Nonconservative forces enter the Lagrange-d’Alembert-Pontryagin principle as virtual work, providing a way to include dissipative effects in the drift.The paper gives standard Langevin equations for particles as an example of such drift effects.Source: §3 · p. 13

Lagrangian Reduction

  • For Lie-group systems, the left-trivialized description uses body velocity ξ=g^-1v and body angular momentum µ∈g*.The associated variational principle is the left-trivialized HP principle.Source: §3 · p. 13

4 Stochastic Variational Integrators

The section expresses stochastic Hamilton–Pontryagin dynamics on Lie groups in left-trivialized form, then derives discrete stochastic Euler–Lagrange updates from a stochastic variational principle.

4 Stochastic Variational Integrators

  • Eliminating xi yields an SDE on G × g*, with the kinematic constraint serving as the reconstruction equation.Source: §4 · p. 14
  • Theorem 3.7 formulates the stochastic left-trivialized HP principle for a Lie-group system with left-trivialized Lagrangian and stochastic potentials.Its action is left-trivialized.Source: §4 · p. 14
  • Almost surely, the stochastic HP principle on a Lie group is equivalent to its left-trivialized form, whose action’s critical points satisfy the left-trivialized HP equations.The critical-point characterization holds with fixed endpoints on the group.Source: §4 · p. 14
  • The discrete stochastic Hamilton principle extremizes the discrete action over paths with fixed endpoints, and stationarity yields stochastic discrete Euler–Lagrange equations.The discrete Lagrangian depends on consecutive configurations and approximates the Stratonovich integrals.Source: §4 · pp. 14–15
  • The resulting update is not self-starting: it requires initial configurations (q0, q1), rather than (q0, v0).Source: §4 · p. 15
  • For symplecticity analysis, restricting the discrete action to pathwise unique solutions leaves only endpoint terms in its first variation.Source: §4 · p. 15

5 Stochastic Variational Euler Integrator

The section extends variational Euler methods to stochastic discrete Hamilton–Pontryagin principles, including Lie-group reconstruction. It establishes symplectic preservation and momentum-map preservation under symmetry.

5 Stochastic Variational Euler Integrator

  • The discrete flow preserves the symplectic form.The argument uses that the second exterior derivative of the discrete generating function is zero.Source: §5 · p. 16
  • If the stochastic discrete Lagrangian is infinitesimally symmetric, its associated momentum map is preserved; left-action invariance is sufficient.The left and right discrete momentum maps then agree.Source: §5 · p. 16
  • A stochastic discrete Hamilton–Pontryagin principle extends variational Euler methods to the stochastic setting.The section examines these methods following the continuous stochastic Hamilton–Pontryagin theory.Source: §5 · p. 16
  • On Lie groups, canonical coordinates of the first kind define a finite-difference map that approximates reconstruction to first order.The map takes G × G to G × g.Source: §5 · p. 17

6 Langevin-type equations for Multiple Rigid Bodies

The section applies the stochastic variational framework to rigid bodies interacting through a pairwise position-and-orientation potential. It describes how constraints, forces, and dissipation enter the resulting Langevin-type systems.

6 Langevin-type equations for Multiple Rigid Bodies

  • Holonomic constraints enter through discrete Lagrange multipliers, while nonconservative effects enter through discrete impulses.For G = Rn, the algorithm is the stochastic analog of constrained symplectic Euler.Source: §6 · p. 18
  • The model comprises K rigid bodies with translational and rotational motion, interacting through a pairwise potential dependent on positions and orientations.Each body's state includes position, velocity, orientation, and spatial angular velocity.Source: §6 · p. 18
  • Extremizing the Hamilton–Pontryagin action with fixed endpoint configurations yields the corresponding stochastic equations of motion.The endpoints fix each body's position and orientation.Source: §6 · p. 18
  • Adding dissipation so the Gibbs distribution remains invariant yields Langevin-type equations for rigid-body systems.Source: §6 · p. 19

Stochastic Variational Euler integrator

Stationarity of the discrete action sum yields a stochastic variational Euler scheme whose configuration updates and translational momentum update are explicit, leaving only the discrete Lie–Poisson part implicit.

Stochastic Variational Euler integrator

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