A Geometrical Approach to Time-Dependent Gauge-Fixing

TL;DR

This paper introduces a geometric method to handle time-dependent constraints in Hamiltonian systems, providing a clear condition for reducing such systems to their physical phase space with a well-defined Hamiltonian evolution.

Contribution

It presents a novel geometric framework that explicitly characterizes when and how a Hamiltonian system with time-dependent constraints can be reduced to its physical phase space.

Findings

Derived an explicit necessary and sufficient condition for reduction

Provided a formula for the Hamiltonian function in constrained systems

Recovered previous results as special cases

Abstract

When a Hamiltonian system is subject to constraints which depend explicitly on time, difficulties can arise in attempting to reduce the system to its physical phase space. Specifically, it is non-trivial to restrict the system in such a way that one can find a Hamiltonian time-evolution equation involving the Dirac bracket. Using a geometrical formulation, we derive an explicit condition which is both necessary and sufficient for this to be possible, and we give a formula defining the resulting Hamiltonian function. Some previous results are recovered as special cases.