Ivan Polekhin

Gubkina str. 8, Moscow, Russia
Steklov Mathematical Institute


Polekhin I. Y.
The Method of Averaging for the Kapitza – Whitney Pendulum
2020, vol. 25, no. 4, pp.  401-410
A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i.e., the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.
Keywords: averaging, Kapitza’s pendulum, Whitney’s pendulum, forced oscillations, averaging on an infinite interval
Citation: Polekhin I. Y.,  The Method of Averaging for the Kapitza – Whitney Pendulum, Regular and Chaotic Dynamics, 2020, vol. 25, no. 4, pp. 401-410
Polekhin I. Y.
The change of the precession angle is studied analytically and numerically for two classical integrable tops: the Kovalevskaya top and the Goryachev – Chaplygin top. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be estimated asymptotically. Some more difficult cases are studied numerically. In particular, we show that the average change of the precession angle for the Kovalevskaya top can be non-zero even in the case of zero area integral.
Keywords: mean motion, Kovalevskaya top, Goryachev – Chaplygin top, integrable system, precession
Citation: Polekhin I. Y.,  Precession of the Kovalevskaya and Goryachev – Chaplygin Tops, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 281-297
Polekhin I. Y.
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
Citation: Polekhin I. Y.,  Classical Perturbation Theory and Resonances in Some Rigid Body Systems, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 136-147

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