On types of stability in Hamiltonian systems

We consider conditions of the formal stability of a stationary solution and conditions of the formal orbital stability of a periodic solution in a Hamiltonian system with finite number of degrees of freedom. The conditions contain restrictions on the order of resonances and some inequalities for initial coefficients of the normal forms of the Hamiltonian functions. We show, that the number-theoretic analysis of frequencies can help in proofs of stability. We estimate also orders of solutions’ divergence from the stationary or the periodic ones under lack of the formal stability.