Symmetries in Constrained Systems

We describe symmetry structure of a general singular theory (theory with constraints in the Hamiltonian formulation), and, in particular, we relate the structure of gauge transformations with the constraint structure. We show that any symmetry transformation can be represented as a sum of three kinds of symmetries: global, gauge, and trivial symmetries. We construct explicitly all the corresponding conserved charges as decompositions in a special constraint basis. The global part of a symmetry does not vanish on the extremals, and the corresponding charge does not vanish on the extremals as well. The gauge part of a symmetry does not vanish on the extremals, but the gauge charge vanishes on them. We stress that the gauge charge necessarily contains a part that vanishes linearly in the first-class constraints and the remaining part of the gauge charge vanishes quadratically on the extremals. The trivial part of any symmetry vanishes on the extremals, and the corresponding charge vanishes quadratically on the extremals.Comment: The talk on Conference "Lie and Jordan algebras, their Representations and Applications II", Brazil, Guaruja, 3-8 May 2004, 9 pages, LaTex fil

Constraint Reorganization Consistent with the Dirac Procedure

The way of finding all the constraints in the Hamiltonian formulation of singular (in particular, gauge) theories is called the Dirac procedure. The constraints are naturally classified according to the correspondig stages of this procedure. On the other hand, it is convenient to reorganize the constraints such that they are explicitly decomposed into the first-class and second-class constraints. We discuss the reorganization of the constraints into the first- and second-class constraints that is consistent with the Dirac procedure, i.e., that does not violate the decomposition of the constraints according to the stages of the Dirac procedure. The possibility of such a reorganization is important for the study of gauge symmetries in the Lagrangian and Hamiltonian formulations.Comment: 18 pages, LaTex fil

Canonical form of Euler-Lagrange equations and gauge symmetries

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.Comment: 27 pages, LaTex fil

Two-dimensional metric and tetrad gravities as constrained second order systems

Using the Gitman-Lyakhovich-Tyutin generalization of the Ostrogradsky method for analyzing singular systems, we consider the Hamiltonian formulation of metric and tetrad gravities in two-dimensional Riemannian spacetime treating them as constrained higher-derivative theories. The algebraic structure of the Poisson brackets of the constraints and the corresponding gauge transformations are investigated in both cases.Comment: replaced with revised version published in Mod.Phys.Lett.A22:17-28,200