Formulation of a constrained system in terms of extended Lagrangian and its local symmetries

TL;DR

This paper presents a method to reformulate any singular Lagrangian theory with constraints into a form with at most third-stage constraints, providing explicit algebraic construction and local symmetries.

Contribution

It introduces a purely algebraic approach to reformulate singular Lagrangian theories, revealing local symmetries and simplifying the constraint structure.

Findings

Reformulation limits constraints to third stage.

Explicit algebraic form of the new Lagrangian $ ilde L$.

First class constraints generate gauge symmetries.

Abstract

It is shown that an arbitrary singular Lagrangian theory (with first and second class constraints up to $N$-th stage in the Hamiltonian formulation) can be reformulated as a theory with at most third-stage constraints. The corresponding Lagrangian $\tilde L$ can be obtained by pure algebraic methods, its manifest form in terms of quantities of the initial formulation is found. Local symmetries of $\tilde L$ are obtained in closed form. All the first class constraints of the initial Lagrangian turn out to be gauge symmetry generators for $\tilde L$.