Ivan Polekhin
Publications:
Polekhin I. Y.
The Method of Averaging for the Kapitza – Whitney Pendulum
2020, vol. 25, no. 4, pp. 401410
Abstract
A generalization of the classical Kapitza pendulum is considered: an inverted planar
mathematical pendulum with a vertically vibrating pivot point in a timeperiodic 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.

Polekhin I. Y.
Precession of the Kovalevskaya and Goryachev – Chaplygin Tops
2019, vol. 24, no. 3, pp. 281297
Abstract
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 nonzero even in the case of zero area integral.

Polekhin I. Y.
Classical Perturbation Theory and Resonances in Some Rigid Body Systems
2017, vol. 22, no. 2, pp. 136147
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.
