On the orbital stability of pendulum oscillations of a heavy rigid body at fourth-order resonance in the case of degeneracy

Аuthors

Maksimov B. A.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: badmamaksimov1@gmail.com

Abstract

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the principal moments of inertia of the body for the fixed point satisfy the D.N. Goryachev – S.A. Chaplygin condition, i.e. are in the ratio  . Unlike the integrable case of D.N. Goryachev – S.A. Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. In the case under consideration, pendulum oscillations of the body relative to the main axis located in the equatorial plane of the ellipsoid of inertia are possible.
The aim of this work is to solve the problem of orbital stability of pendulum oscillations of a body in previously unexplored cases corresponding to fourth-order resonance in the presence of degeneracy, when to obtain rigorous conclusions about orbital stability it is necessary to conduct an analysis up to sixth-order terms inclusive in the expansion of the Hamiltonian function in the neighborhood of unperturbed periodic motion. The study of resonance cases is of both theoretical interest and is important for applications, since it allows us to identify the parameter values at which a qualitative change in the nature of the motion of systems occurs. The study is based on the method of normal forms and the KAM theory. This made it possible to obtain rigorous conclusions about orbital instability, as well as conclusions about orbital stability at the energy level corresponding to the unperturbed orbit. The results of the study are presented in the table and illustrated in the stability diagram.

Keywords:

pendulum oscillations, orbital stability, the case of D.N. Goryachev – S.A. Chaplygin, fourth-order resonance, local variables, Hamiltonian systems.

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