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Stochastic Implicit Lagrange-Poincaré Reduction

Archishman Saha

TL;DR

The paper tackles the challenge of incorporating symmetry into stochastic variational principles by extending the implicit Lagrange-Poincaré reduction to stochastic settings using the stochastic Hamilton-Pontryagin framework. The authors develop a reduced action on $\mathcal{P}(Q/G)\oplus \tilde{\mathfrak{g}}\oplus \tilde{\mathfrak{g}}^*$ and derive stochastic horizontal and vertical Lagrange-Poincaré equations that couple curvature and noise through the stochastic covariant derivative. The approach relies on covariant Stratonovich calculus to preserve coordinate independence and provides a principled variational basis for reduced stochastic dynamics, demonstrated on a stochastic rigid body with rotor and a Kaluza–Klein model of a charged particle in a magnetic field. Overall, the work clarifies how curvature and symmetry interact with stochastic perturbations in reduced mechanical systems and offers a framework for deriving reduced stochastic equations with rigorous geometric structure.

Abstract

In this paper we consider reduction of the stochastic Hamilton-Pontryagin principle formulated on the Pontryagin bundle of a manifold $Q$. We prove that a stochastic action invariant under the free and proper action of a Lie group $G$ drops to a reduced variational principle expressed in terms of variables of the Pontryagin bundle of the reduced space $Q/G$, the associated adjoint bundle $\tilde{\mathfrak{g}}:= (Q\times \mathfrak{g})/G$ and its dual bundle $\tilde{\mathfrak{g}}^*$. This provides a stochastic analogue of the deterministic implicit Lagrange-Poincaré reduction. The stochastic Euler-Lagrange equations drop to a set of stochastic horizontal and vertical Lagrange-Poincaré equations on $T(Q/G)\oplus T^*(Q/G)\oplus\tilde{\mathfrak{g}}\oplus\tilde{\mathfrak{g}}^*$. As examples, we consider stochastic perturbations of the rigid body with a rotor, as well as a Kaluza-Klein description of stochastic perturbations of a charged particle in a magnetic field.

Stochastic Implicit Lagrange-Poincaré Reduction

TL;DR

The paper tackles the challenge of incorporating symmetry into stochastic variational principles by extending the implicit Lagrange-Poincaré reduction to stochastic settings using the stochastic Hamilton-Pontryagin framework. The authors develop a reduced action on and derive stochastic horizontal and vertical Lagrange-Poincaré equations that couple curvature and noise through the stochastic covariant derivative. The approach relies on covariant Stratonovich calculus to preserve coordinate independence and provides a principled variational basis for reduced stochastic dynamics, demonstrated on a stochastic rigid body with rotor and a Kaluza–Klein model of a charged particle in a magnetic field. Overall, the work clarifies how curvature and symmetry interact with stochastic perturbations in reduced mechanical systems and offers a framework for deriving reduced stochastic equations with rigorous geometric structure.

Abstract

In this paper we consider reduction of the stochastic Hamilton-Pontryagin principle formulated on the Pontryagin bundle of a manifold . We prove that a stochastic action invariant under the free and proper action of a Lie group drops to a reduced variational principle expressed in terms of variables of the Pontryagin bundle of the reduced space , the associated adjoint bundle and its dual bundle . This provides a stochastic analogue of the deterministic implicit Lagrange-Poincaré reduction. The stochastic Euler-Lagrange equations drop to a set of stochastic horizontal and vertical Lagrange-Poincaré equations on . As examples, we consider stochastic perturbations of the rigid body with a rotor, as well as a Kaluza-Klein description of stochastic perturbations of a charged particle in a magnetic field.
Paper Structure (21 sections, 16 theorems, 127 equations)

This paper contains 21 sections, 16 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Implicit Lagrange-Poincaré Reduction
  3. The Hamilton-Pontryagin Principle
  4. The Geometric Setup of Implicit Lagrange-Poincaré Reduction
  5. The Structure of Variations
  6. The Implicit Lagrange-Poincaré Reduction Theorem
  7. The Stochastic Hamilton-Pontryagin Principle
  8. Deformations of Manifold-Valued Semimartingale
  9. The Stochastic Implicit Euler-Lagrange Equations
  10. Connectors and the Stochastic Covariant Derivative
  11. The Connector on a Vector Bundle
  12. The Stochastic Covariant Derivative
  13. Stochastic Implicit Lagrange-Poincaré Reduction
  14. The Reduced Action Functional
  15. Vertical and Horizontal Variations
  16. ...and 6 more sections

Key Result

Theorem 2.1

A curve $(q(t), v(t), p(t))$ in $\mathcal{P} Q$ is a critical point for $\mathcal{S}$ for deformations $\epsilon\mapsto (q_{\epsilon}(t), v_{\epsilon}(t), p_{\epsilon}(t))$ such that $\delta q(t) = 0$ at $t = 0$ and $t = T$ if and only if $(q(t), v(t), p(t))$ satisfies the implicit Euler-Lagrange eq

Theorems & Definitions (41)

  • Theorem 2.1
  • Lemma 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3: Implicit Lagrange-Poincaré Reduction Theorem
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.1
  • ...and 31 more