Regular and Chaotic Dynamics

, Volume 22, Issue 2, pp 136–147 | Cite as

Classical perturbation theory and resonances in some rigid body systems

  • Ivan Yu. PolekhinEmail author


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.


Poincaré method Poincaré sets resonances periodic solutions small divisors rigid body Kirchhoff’s equations 

MSC2010 numbers

70E17 70E20 70E40 


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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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