Recursive Construction of Generator for Lagrangian Gauge Symmetries

We obtain, for a subclass of structure functions characterizing a first class Hamiltonian system, recursive relations from which the general form of the local symmetry transformations can be constructed in terms of the independent gauge parameters. We apply this to a nontrivial Hamiltonian system involving two primary constraints, as well as two secondary constraints of the Nambu-Goto type. email: [email protected] On leave of absence from S.N. Bose Natl. Ctr. for Basic Sc., Salt Lake, Calcutta 700091, India email: [email protected] email: [email protected] The problem of finding the most general local symmetries of a Lagrangian has been pursued by various authors, using either Lagrangian [1, 2, 3, 4] or Hamiltonian techniques [5, 6, 7, 8]. In a recent paper [9] we had shown that the requirement of commutativity of the time derivative operation with an arbitrary infinitesimal gauge variation generated by the first class constraints was the only input needed for obtaining the restrictions on the gauge parameters entering the most general form of the generator of Lagrangian symmetries. The analysis was performed entirely in the Hamiltonian framework. On the basis of this commutativity requirement, we subsequently derived [10] a simple differential equation for the generator encoding, in particular, the restrictions on the gauge parameters. In this paper we shall obtain, for a subclass of structure functions characterizing a first class Hamiltonian system, the explicit solution of the above differential equation in the form of simple recursive relations. We then apply this general scheme to a non-trivial model discussed in the literature [12], whose secondary (first class) constraints are identical with the primary constraints of the Nambu-Goto model. Our result for the gauge transformation is found to agree with that quoted in the literature. We shall consider purely first class systems. The extension to mixed first and second class systems is straightforward. To keep the algebra simple we assume all constraints to be irreducible. Consider a Hamiltonian system whose dynamics is described by the total Hamiltonian 4 HT = Hc + ∑ a1 vΦa1 . (1) where Hc is the canonical Hamiltonian, {Φa1 ≈ 0} are the (first class) primary constraints, and v1 are the associated Lagrange multipliers. We denote the We follow here the notation of Ref. [9].