On the Symmetries of Hamiltonian Systems

TL;DR

This paper explores how local symmetries, especially diffeomorphism invariance, manifest in Hamiltonian systems and distinguishes between linear and nonlinear constraints in generating symmetries versus dynamics.

Contribution

It demonstrates that only linear constraints generate symmetries, while nonlinear constraints relate to system dynamics, and establishes the connection between Hamiltonian and Lagrangian symmetry parameters.

Findings

Linear constraints generate system symmetries.

Nonlinear constraints primarily generate dynamics.

Special trivial transformations relate symmetries to equations of motion.

Abstract

In this paper we show how the well-know local symmetries of Lagrangeans systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian- and Lagrangean formalisms is found. The possible applications of our results are discussed.