Hamilton-Jacobi theory, symmetries and coisotropic reduction

Reduction theory has played a major role in the study of Hamiltonian systems. Whilst 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 relies 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 reduction of the Hamilton-Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge-Marsden reduction procedure [18] and the results in [17]. Quite surprisingly, the classical ansatze available in the literature to solve the Hamilton -Jacobi equation (see [2, 19]) are also particular instances of our framework. (C) 2016 Elsevier Masson SAS. All rights reserved.