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arXiv · 1509.00419

Hamilton-Jacobi theory, Symmetries and Coisotropic Reduction

Manuel de León·David Martín de Diego·Miguel Vaquero
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Abstract

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approximations of a complete solution of the Hamilton-Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton-Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton-Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reductions of the Hamilton-Jacobi theory are shown in the last section of the paper. It is remarkable that as a by-product we obtain a generalization of the Ge-Marsden reduction procedure. Quite surprinsingly, the classical ansatzs available in the literature to solve the Hamilton-Jacobi equation are also particular instances of our framework.

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Manuel de León, David Martín de Diego, Miguel Vaquero. 2015-09-01. Hamilton-Jacobi theory, Symmetries and Coisotropic Reduction. https://arxiv.org/abs/1509.00419

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