Existence of isotropic complete solutions of the P-Hamilton-Jacobi Equation

SERGIO GRILLO

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2020

0393-0440

Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration P: M -> N. Related to thesedata we have a generalized version of the (time-independent) Hamilton-Jacobi equation: theP-HJE for X, whoseunknown is a section ofP. The standard HJE is obtained when the phase space M is a cotangent bundle(with its canonical symplectic form), P is the canonical projection and the unknown is a closed1-form dW. The function W is called Hamilton´s characteristic function. Coming back to the generalizedversion, among the solutions of the P-HJE, a central role is played by the so-called isotropic complete solutions. Thisis because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integratedup to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic completesolution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative prooffor the local existence of a complete family of Hamilton´s characteristic functions.