Notes on Lagrangean and Hamiltonian Symmetries

The Hamiltonization of local symmetries of the form $\delta q^A = \ea{R_a}^A(q,\dot q)$ or $\delta q^A = \dot\ea{R_a}^A (q,\dot q)$ for arbitrary Lagrangean model $L(q^A,\dot q^A)$ is considered. We show as the initial symmetries are transformed in the transition from $L$ to first order action, and then to the Hamiltonian action $S_H=\int{\rm d}\tau(p_A\dot q^A-H_0-v^\alpha\Phi_\alpha)$, where $\Phi_\alpha$ are the all (first and second class) primary constraints. An exact formulae for local symmetries of $S_H$ in terms of the initial generators ${R_a}^A$ and all primary constraints $\Phi_\alpha$ are obtained.