Rigid and gauge noether symmetries for constrained systems

We develop the general theory of Noether symmetries for constrained systems , that is, systems that are described by singular Lagrangians. In our deriv ation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of th e conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fu lly analyzed, and we give a geometrical interpretation of the projectabilit y conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We also examine the enlarged formalism that results from taking the Lagrange multipliers as new dynamical variables; we find the equation that characterizes the Noeth er symmetries in this formalism, and we also prove that the standard formul ation is a particular case of the enlarged one. The algebra of generators f or Noether symmetries is discussed in both the Hamiltonian and Lagrangian f ormalisms. We find that a frequent source for the appearance of open algebr as is the fact that the transformations of momenta in phase space and tange nt space only coincide on shell. Our results apply with no distinction to r igid and gauge symmetries; for the latter case are give a general proof of the existence of Noether gauge symmetries for theories with first and secon d class constraints that do not exhibit tertiary constraints in the stabili zation algorithm. Among some examples that illustrate our results, we study the Noether gauge symmetries of the Abelian Chern-Simons theory in 2n + 1 dimensions. An interesting feature of this example is that its primary firs t class constraints can only be identified after the determination of the s econdary constraint. The example is worked out retaining all the original s et of variables.