Structure of Constrained Systems in Lagrangian Formalism and Degree of Freedom Count

Abstract

A detailed program is proposed in the Lagrangian formalism to investigate the dynamical behavior of a theory with singular Lagrangian. This program goes on, at different levels, parallel to the Hamiltonian analysis. In particular, we introduce the notions of first class and second class Lagrangian constraints. We show each sequence of first class constraints leads to a Neother identity and consequently to a gauge transformation. We give a general formula for counting the dynamical variables in Lagrangian formalism. As the main advantage of Lagrangian approach, we show the whole procedure can also be performed covariantly. Several examples are given to make our Lagrangian approach clear.

Lagrangian constraint analysis of first-order classical field theories with an application to gravity

“…This is because it leads to a secondary constraint (3.5) that is identically satisfied and so non-dynamically stabilized by means of the closure iia. In fact, this same example is worked out in [4] as well.…”

“…During the preparation of this manuscript, a novel Lagrangian approach to obtain the functionally independent Lagrangian constraints and count propagating modes in constrained systems (of the kind here considered) appeared [4]. The method therein is physically equivalent to that put forward in [2,3], which -as already mentioned-are complementary references to our own discussion in section 2.…”

“…Our approach is complementary to the existing literature. In particular, it is equivalent to the recent proposal in [4], as argued and exemplified in section 5.1.…”

“…In our understanding, the method in [4] distinguishes itself because it introduces the notion of first and second class (functionally independent) Lagrangian constraints. In our language, these are easy to identify.…”

“…To do so, we explicitly obtain the constraints: specific functional relations among the field variables and their time derivatives that avoid the propagation of the remaining field components. Our approach is complementary to the similarly aimed procedures in [1][2][3] and is markedly distinct from, yet equivalent to, that in [4].…”