Abstract
Periodic solutions of a canonical system of differential equations with a special type Hamiltonian, i.e. of the so-called fundamental problem of dynamics /1/, are investigated. A method of constructing the conditions of periodicity of the solutions is given and a non-linear analysis of the solutions is carried out. The method enables the Poincare's classical conditions of existence, as well as of the new conditions of existence of periodic solutions in degenerate cases to be derived. The cases of degeneracy discussed here appear very frequently in various problems of dynamics. The results obtained are illustrated by finding new periodic solutions for the problem of the motion of a heavy rigid body about a fixed point.
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