ON THE SYMMETRIES OF HAMILTONIAN-SYSTEMS

In this paper we show how the well-known local symmetries of Lagrangia n systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which a re linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The nonlinear const raints (which we have, for instance, in gravity, supergravity and stri ng theory) generate the dynamics of the corresponding Lagrangian syste m. Only in a very special combination with ''trivial'' transformations proportional to the equations of motion do they lead to symmetry tran sformations. We show the importance of these special ''trivial'' trans formations for the interconnection theorems which relate the symmetrie s of a system with its dynamics. We prove these theorems for general H amiltonian systems. We apply the developed formalism to concrete physi cally relevant systems, in particular those which are diffeomorphism-i nvariant. The connection between the parameters of the symmetry transf ormations in the Hamiltonian and Lagrangian formalisms is found. The p ossible applications of our results are discussed.