A Hamilton-Jacobi theory for implicit differential systems

TL;DR

This paper develops a geometric Hamilton-Jacobi framework for implicit differential systems, especially implicit Hamiltonian systems, using Morse families and Lagrangian submanifolds, addressing the challenge of absent Hamiltonian functions.

Contribution

It introduces a Hamilton-Jacobi theory for implicit systems via Morse families, extending classical methods to systems without explicit Hamiltonians.

Findings

Formulation of a Hamilton-Jacobi equation for implicit systems.

Application to singular Lagrangians with special symplectic structures.

Use of Morse families to generate Hamiltonian dynamics in implicit systems.

Abstract

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $TT^*Q$ generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton--Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.