Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

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 ξ t , 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 ξ t is valid for all ξ inside an open dense subset of g .