Lagrangean Approach to Hamiltonian Gauge Symmetries and the Dirac Conjecture

TL;DR

This paper demonstrates how Lagrangian techniques can be used to derive gauge symmetries in Hamiltonian systems, confirming the Dirac conjecture and clarifying the connection between Lagrangian and Hamiltonian approaches.

Contribution

It provides a method to derive gauge transformation laws from Lagrangian principles that align with Hamiltonian results based on the Dirac conjecture.

Findings

Lagrangian techniques reproduce Hamiltonian gauge transformations.

The approach confirms the Dirac conjecture for first class systems.

Illustration with systems involving one primary constraint.

Abstract

Using well known Lagrangean techniques for uncovering the gauge symmetries of a Lagrangean, we derive the transformation laws for the phase space variables corresponding to local symmetries of the Hamilton equations of motion. These transformation laws are shown to coincide with those derived by Hamiltonian methods based on the Dirac conjecture. The connection between the Lagrangean and Hamiltonian approach is illustrated for first class systems involving one primary constraint.