  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.
