Canonical form of Euler-Lagrange equations and gauge symmetries

TL;DR

This paper develops a method to transform Euler-Lagrange equations into a canonical form, clarifying gauge symmetries and constraints in singular Lagrangian systems, and providing a systematic way to identify gauge generators.

Contribution

It introduces a reduction procedure to canonical form for Euler-Lagrange equations, revealing gauge identities and generators within the Lagrangian framework.

Findings

Reduction procedure reveals constraints similar to Dirac's method

Constructive identification of all gauge generators in Lagrangian form

Proves gauge generators are local in time for local theories

Abstract

The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.