Orbital stability of the periodic solution of a Hamiltonian system

In a vicinity of a periodic solution of an autonomous Hamiltonian system we introduce local canonical coordinates. Then we make a formal canonical transformation of the coordinates, reducing the Hamiltonian function to the complex normal form. Next we make more precision properties of the normal form in real case and, using coefficients of the beginning terms of the normal form, we give conditions, which are sufficient for the formal orbital stability of the initial periodic solution. We give also the corresponding proof. By means of counterexamples we show that the A.P. Markeev’s conditions of the stability are wrong. So results of their applications in mechanical problems by A.P. Markeev and B.S. Bardin should be revised.