Improved extended Hamiltonian and search for local symmetries

TL;DR

This paper introduces an extended Lagrangian formalism that simplifies the analysis of local symmetries and constraints in singular Lagrangian systems, confirming the Dirac conjecture within the Lagrangian framework.

Contribution

The authors develop a closed-form extended Lagrangian that generates all original constraints and simplifies the derivation of gauge generators, confirming the Dirac conjecture.

Findings

Extended Lagrangian generates all original constraints.

Gauge generators are directly expressed through first class constraints.

The formalism confirms the Dirac conjecture in the Lagrangian approach.

Abstract

We analyze a structure of the singular Lagrangian $L$ with first and second class constraints of an arbitrary stage. We show that there exist an equivalent Lagrangian (called the extended Lagrangian $\tilde L$) that generates all the original constraints on second stage of the Dirac-Bergmann procedure. The extended Lagrangian is obtained in closed form through the initial one. The formalism implies an extension of the original configuration space by auxiliary variables. Some of them are identified with gauge fields supplying local symmetries of $\tilde L$. As an application of the formalism, we found closed expression for the gauge generators of $\tilde L$ through the first class constraints. It turns out to be much more easy task as those for $L$. All the first class constraints of $L$ turn out to be the gauge symmetry generators of $\tilde L$. By this way, local symmetries of $L$ with higher order derivatives of the local parameters decompose into a sum of the gauge symmetries of $\tilde L$. It proves the Dirac conjecture in the Lagrangian framework.