Hamilton-Jacobi theory, symmetries and coisotropic reduction

Title: Hamilton-Jacobi theory, symmetries and coisotropic reduction.

Author(s): Manuel de León, David Martín de Diego and Miguel Vaquero.

Source: J. Math. Pures Appl. in press.

Date of publication: August 1, 2016.

Abstract: 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 ansätze available in the literature to solve the Hamilton-Jacobi equation (see [2] and [19]) are also particular instances of our framework.

Link: http://www.sciencedirect.com/science/article/pii/S0021782416300836