Volume 24, Number 3
Volume 24, Number 3, 2019
Kholostova O. V.
On the Motions of One NearAutonomous Hamiltonian System at a 1:1:1 Resonance
Abstract
We consider the motion of a $ 2 \pi $periodic in time twodegreeoffreedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter $ e $ and other parameters and is autonomous at $ e = 0 $. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced twodegreeoffreedom system in the vicinity of the resonance point in the threedimensional parameter space is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of $ 2 \pi$periodic motions of the satellite is solved, and conclusions on the existence of two and threefrequency conditionally periodic motions are obtained.

Bolsinov A. V., Bao J.
A Note about Integrable Systems on Lowdimensional Lie Groups and Lie Algebras
Abstract
The goal of the paper is to explain why any leftinvariant Hamiltonian system on (the cotangent bundle of) a 3dimensonal Lie group $G$ is Liouville integrable. We derive this property from the fact that the coadjoint orbits of $G$ are twodimensional so that the integrability of leftinvariant systems is a common property of all such groups regardless their dimension.
We also give normal forms for leftinvariant Riemannian and subRiemannian metrics on 3dimensional Lie groups focusing on the case of solvable groups, as the cases of $SO(3)$ and $SL(2)$ have been already extensively studied. Our description is explicit and is given in global coordinates on $G$ which allows one to easily obtain parametric equations of geodesics in quadratures. 
Polekhin I. Y.
Precession of the Kovalevskaya and Goryachev – Chaplygin Tops
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.

Pepa R. Y., Popelensky T. Y.
Combinatorial Ricci Flow for Degenerate Circle Packing Metrics
Abstract
Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence can be proved under a weaker condition: some weights can be negative and should satisfy certain inequalities [4].
On the other hand, for weights not satisfying conditions of Chow – Luo’s theorem we observed in numerical simulation a degeneration of the metric with certain regular behaviour patterns [5]. In this note we introduce degenerate circle packing metrics, and under weakened conditions on weights we prove that under certain assumptions for any initial metric an analogue of the combinatorial Ricci flow has a unique limit metric with a constant curvature outside of singularities. 
Ardentov A. A., Karavaev Y. L., Yefremov K. S.
Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization
Abstract
This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice are discussed. In particular, analysis is made of speedup and deceleration along the elastica, and of the influence of the errors made in manufacturing the mobile robot on the precision with which the prescribed trajectory is followed. Special attention is also given to the problem of forming optimal trajectories of motion along Euler elasticas to a preset point at different angles of orientation. Results of experimental investigations are presented.

Borisov A. V., Kilin A. A., Mamaev I. S.
A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness
Abstract
This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both wellknown results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of threedimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point. 