Аннотация:
We deal with the problem of orbital stability of the periodic solutions of a Hamiltonian
system with two degrees of freedom. We assume that third- or sixth-order resonance takes place
in this system and solve the orbital stability problem in a special case of degeneracy, when it is
necessary to take into account the terms up to the sixth order of the Hamiltonian expansion in
the neighborhood of the periodic solution. By using methods of normal forms and KAM theory
we obtain sufficient conditions of orbital stability and instability in the form of inequalities with
respect to coefficients of the Hamiltonian normal form calculated up to terms of the sixth order.
We also show that the above-mentioned problem of orbital stability is equivalent to the problem
of Lyapunov stability of an equilibrium position of a reduced system.
We apply the above results in the problem of the orbital stability of the pendulum oscillations of a heavy rigid body with a fixed point in the case when its principal moment of
inertia $A$, $B$, and $C$ satisfy the equality $A = C = 4B$.
Ключевые слова:
Hamiltonian system, normal form, orbital stability, resonance, rigid body oscillations
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