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 Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 4, Pages 581–594 (Mi nd730)

Nonlinear physics and mechanics

On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body

B. S. Bardinab

a Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia
b Mechanical Engineering Research Institute of the Russian Academy of Sciences, M. Kharitonyevskiy per. 4, Moscow, 101990 Russia

Abstract: A method is presented of constructing a nonlinear canonical change of variables which makes it possible to introduce local coordinates in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. The problem of the orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev – Steklov case is discussed as an application. The nonlinear analysis of orbital stability is carried out including terms through degree six in the expansion of the Hamiltonian function in a neighborhood of the unperturbed periodic motion. This makes it possible to draw rigorous conclusions on orbital stability for the parameter values corresponding to degeneracy of terms of degree four in the normal form of the Hamiltonian function of equations of perturbed motion.

Keywords: rigid body, rotations, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form

 Funding Agency Grant Number Russian Foundation for Basic Research 20-01-00637 This work was supported by the Russian Foundation for Basic Research, project No. 20-01-00637.

DOI: https://doi.org/10.20537/nd200404

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MSC: 34D20, 37J40, 70K30, 70K45, 37N05
Accepted:25.12.2020

Citation: B. S. Bardin, “On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body”, Rus. J. Nonlin. Dyn., 16:4 (2020), 581–594

Citation in format AMSBIB
\Bibitem{Bar20} \by B. S. Bardin \paper On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body \jour Rus. J. Nonlin. Dyn. \yr 2020 \vol 16 \issue 4 \pages 581--594 \mathnet{http://mi.mathnet.ru/nd730} \crossref{https://doi.org/10.20537/nd200404} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4198781}