A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds

Abstract

In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton-Jacobi theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton-Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is stablished. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.

Existence of isotropic complete solutions of the Π-Hamilton–Jacobi equation

“…To show our main result, we shall use the duality between (isotropic) complete solutions and (isotropic) first integrals stablished in [12]. Let us recall such a duality for the case of Hamiltonian systems.…”

“…Then, if X H (m) / ∈ Ker (π Q ) * ,m , above theorem ensures that a local isotropic complete solution exists around m. But, since k = s, such a solution is actually Lagrangian. As explained in the Introduction (see [12] for more details), the Lagrangian complete solutions of the π Q -HJE for H are locally given by the expression…”

“…In Ref. [12], it was presented an extension to general dynamical systems (on fibered phase spaces), which contains as particular cases the previuos ones. Let us briefly review it in the restricted context of Hamiltonian systems.…”

“…More precisely (see Ref. [12]), in this case the complete solutions are locally given by the formula Σ (q, λ) = dW λ (q), where each function W λ , called…”

“…The importance of the isotropic complete solutions, as proven in [11,12] (generalizing a well-known result of the standard theory), lies in that fact that if we know one of them for a given Hamiltonian system, then such a system is exactly solvable. Actually, in order to ensure exact solvability, it is enough to know a local solution (instead of a global one) around every point of the phase space.…”