Non-Involutive Constrained Systems and Hamilton-Jacobi Formalism

TL;DR

This paper explores how generalized brackets naturally arise in non-involutive constrained systems within the Hamilton-Jacobi framework, providing a geometric interpretation that reduces degrees of freedom and defines dynamics in a simplified phase space.

Contribution

It introduces a geometric interpretation of integrability conditions that leads to a natural reduction of degrees of freedom in non-involutive constrained systems.

Findings

Generalized brackets appear naturally in non-involutive systems

A geometric interpretation of integrability conditions is provided

Reduced phase space dynamics are established

Abstract

In this work we discuss the natural appearance of the Generalized Brackets in systems with non-involutive (equivalent to second class) constraints in the Hamilton-Jacobi formalism. We show how a consistent geometric interpretation of the integrability conditions leads to the reduction of degrees of freedom of these systems and, as consequence, naturally defines a dynamics in a reduced phase space.