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Local symmetries in the Hamiltonian framework. 1. Hamiltonian form of the symmetries and the Noether identities

We study in the Hamiltonian framework the local transformations $δ_εq^A(τ)=\sum^{[k]}_{k=0}\partial^k_τε^a{} R_{(k)a}{}^A(q^B, \dot q^C)$ which leave invariant the Lagrangian action: $δ_εS=div$. Manifest form of the symmetry and the corresponding Noether identities is obtained in the first order formalism as well as in the Hamiltonian one. The identities has very simple form and interpretation in the Hamiltonian framework. Part of them allows one to express the symmetry generators which correspond to the primarily expressible velocities through the remaining one. Other part of the identities allows one to select subsystem of constraints with a special structure from the complete constraint system. It means, in particular, that the above written symmetry implies an appearance of the Hamiltonian constraints up to at least $([k]+1)$ stage. It is proved also that the Hamiltonian symmetries can always be presented in the form of canonical transformation for the phase space variables. Manifest form of the resulting generating function is obtained.