Extended Hamilton-Jacobi Theory, Symmetries and Integrability by Quadratures

Abstract

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that π(U) has a manifold structure and π|U:U→π(U), the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If X|U is not vertical with respect to π|U, we show that such complete solutions solve the reconstruction equations related to X|U and G, i.e., the equations that enable us to write the integral curves of X|U in terms of those of its projection on π(U). On the other hand, if X|U is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of X|U up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve exp(ξt), different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of exp(ξt) is valid for all ξ inside an open dense subset of g.