Canonical form of Euler–Lagrange equations and gauge symmetries

Abstract

The structure of the Euler-Lagrange equations for a general Lagrangian theory (e.g. singular, with higher derivatives) 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 EulerLagrange 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.

Darboux Coordinates for the Hamiltonian of First Order Einstein-Cartan Gravity

“…any invertible transformation) known for non-singular Lagrangians without careful analysis and without taking into account the specifics of singular systems (see a general discussion in [29]). When constructing the Darboux coordinates, we rely on the Hamiltonian analysis and this protects us from such mistakes.…”

“…The importance of careful preliminary analysis before doing the Lagrange reduction was emphasized in [29]: "it seems important to develop reduction procedure within Lagrangian formulation -in a sense similar to the Dirac procedure in the Hamiltonian formulation -that may allow one to reveal the hidden structure of the Euler-Lagrange equations of motion in a constructive manner".…”

“…This is the illustration of how a pure Lagrangian consideration of singular systems can destroy its properties if one assumes that one can always use some operations (e.g. any invertible transformation) known for non-singular Lagrangians without careful analysis and without taking into account the specifics of singular systems (see a general discussion in [29]). When constructing the Darboux coordinates, we rely on the Hamiltonian analysis and this protects us from such mistakes.…”

“…However, we have to emphasize that the preliminary Hamiltonian analysis is indispensable for the construction of Darboux coordinates, which preserve equivalence with the original action.An arbitrary change of variables at the Lagrangian level for singular Lagrangians is an ambiguous operation because it might correspond to a non-canonical change of variables at the Hamiltonian level. For singular Lagrangians the invertability of transformations (redefinition of fields) from one set of variables to another is not a sufficient condition to preserve equivalence[29]; and one particular example is considered in the end of Section III (see (53)). These "Darboux coordinates", (53), despite separating variables in the same way, do not preserve the D = 3 limit, lead to results which are different from those found by the direct analysis and destroy equivalence.…”