Classical perturbation theory and resonances in some rigid body systems
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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.
KeywordsPoincaré method Poincaré sets resonances periodic solutions small divisors rigid body Kirchhoff’s equations
MSC2010 numbers70E17 70E20 70E40
- 3.Born, M., The Mechanics of the Atom, New York: Ungar, 1967.Google Scholar
- 5.Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Jerusalem: Israel Program for Scientific Translations, 1960.Google Scholar
- 10.Denisova, N.V., Kozlov, V.V., and Treshchev, D.V., Remarks on Polynomial Integrals of Higher Degree for Reversible Systems with a Toral Configuration Space, Izv. Math., 2012, vol. 76, no. 5, pp. 907–921; see also: Izv. Ross. Akad. Nauk Ser. Mat., 2012, vol. 76, no. 5, pp. 57–72.MathSciNetCrossRefzbMATHGoogle Scholar