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 Regul. Chaotic Dyn., 2017, Volume 22, Issue 2, Pages 136–147 (Mi rcd247)

Classical Perturbation Theory and Resonances in Some Rigid Body Systems

Ivan Yu. Polekhin

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff's equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.

Keywords: Poincaré method, Poincaré sets, resonances, periodic solutions, small divisors, rigid body, Kirchhoff's equations

DOI: https://doi.org/10.1134/S1560354717020034

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MSC: 70E17, 70E20, 70E40
Accepted:15.01.2017
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Citation: Ivan Yu. Polekhin, “Classical Perturbation Theory and Resonances in Some Rigid Body Systems”, Regul. Chaotic Dyn., 22:2 (2017), 136–147

Citation in format AMSBIB
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