Stochastic Implicit Lagrange-Poincaré Reduction
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917202839470080 |
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| author | Saha, Archishman |
| author_facet | Saha, Archishman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08994 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Implicit Lagrange-Poincaré Reduction Saha, Archishman Mathematical Physics Dynamical Systems 70L10, 70H33 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. |
| title | Stochastic Implicit Lagrange-Poincaré Reduction |
| topic | Mathematical Physics Dynamical Systems 70L10, 70H33 |
| url | https://arxiv.org/abs/2601.08994 |