Symmetries of dynamically equivalent theories

TL;DR

This paper proves that the symmetry structures of Hamiltonian and Lagrangian actions are equivalent for constrained systems, establishing an isomorphism between their symmetry classes, which simplifies the analysis of such theories.

Contribution

It demonstrates that the symmetry structures of dynamically equivalent Hamiltonian and Lagrangian actions are isomorphic, clarifying their relationship in constrained system theory.

Findings

Symmetry structures of Hamiltonian and Lagrangian actions are the same.

An isomorphism exists between classes of equivalent symmetries.

The study simplifies the analysis of symmetries in constrained systems.

Abstract

A natural and very important development of constrained system theory is a detail study of the relation between the constraint structure in the Hamiltonian formulation with specific features of the theory in the Lagrangian formulation, especially the relation between the constraint structure with the symmetries of the Lagrangian action. An important preliminary step in this direction is a strict demonstration, and this is the aim of the present article, that the symmetry structures of the Hamiltonian action and of the Lagrangian action are the same. This proved, it is sufficient to consider the symmetry structure of the Hamiltonian action. The latter problem is, in some sense, simpler because the Hamiltonian action is a first-order action. At the same time, the study of the symmetry of the Hamiltonian action naturally involves Hamiltonian constraints as basic objects. One can see that the Lagrangian and Hamiltonian actions are dynamically equivalent. This is why, in the present article, we consider from the very beginning a more general problem: how the symmetry structures of dynamically equivalent actions are related. First, we present some necessary notions and relations concerning infinitesimal symmetries in general, as well as a strict definition of dynamically equivalent actions. Finally, we demonstrate that there exists an isomorphism between classes of equivalent symmetries of dynamically equivalent actions.