Reduction of hybrid Hamiltonian systems with non-equivariant momentum maps

TL;DR

The paper addresses reducing hybrid Hamiltonian systems when the momentum map $J$ is not equivariant under the coadjoint action. It introduces an affine action $\Psi$ with a cocycle $\sigma$ to render $J$ equivariant, enabling a non-equivariant reduction framework via a generalized hybrid momentum map; this leads to a hybrid Marsden--Weinstein--Meyer reduction with reduced spaces $D_{\mu}=J^{-1}(\mu)/\tilde{G}_{\mu}$ that inherit a symplectic form and induce reduced dynamics when the Hamiltonian is invariant. Under suitable regularity and equivariance of the hybrid action, isotropy subgroups for successive regular values coincide, allowing reduction by stages along a sequence of $\mu$'s and ensuring the reduced hybrid systems $\mathcal{H}_{\mu}$ properly reflect the original flow. An explicit example with $Q=\mathbb{R}^2$ demonstrates the construction, showing how $H_{\mu}$ is obtained from $H$ and how the reduced space and guards/readouts are realized. Overall, the work broadens symmetry reduction to hybrid systems with non-equivariant momentum maps, enabling downstream analysis and applications in mechanical systems with impacts.

Abstract

We develop a reduction scheme à la Marsden-Weinstein-Meyer for hybrid Hamiltonian systems. Our method does not require the momentum map to be equivariant, neither to be preserved by the impact map. We illustrate the applicability of our theory with an example.

Theorems & Definitions (5)

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