Theoretical and experimental investigations of the controlled motion of a roller racer
Alexander A. Kilin, Tatiana B. Ivanova, Yury L. Karavaev, and Kirill S. Yefremov
Available online 25 December 2024

Abstract
In this paper, we address the problem of the controlled motion of a roller racer on a plane. We assume that the angle between the platforms is a given periodic function of time (control function), and the no-slip conditions (nonholonomic constraint) and viscous friction forces act at the points of contact of the wheels with the plane. In this case, all trajectories of the reduced system tend asymptotically to a periodic solution. In this paper, we show that for a selected periodic control function there exists a motion of the system that is bounded (along a circle) and unbounded (along a straight line). Unbounded motion corresponds to the resonant case which takes place at zero average value of the control function. The theoretical dependence of the trajectory and the velocity of the roller racer on its parameters and the parameters of the selected control function is investigated. These dependences are confirmed experimentally.

Mathematics Subject Classification
37J60, 70E60

Keywords
roller racer, nonholonomic constraint, viscous friction, control, periodic solution

DOI
https://doi.org/10.2298/TAM241203010K

Experimental observation of stabilization of tippe top spinning on a vibrating plane
Alexander A. Kilin and Yury L. Karavaev
Available online 10 January 2025

Abstract
This paper presents an experimental study of the motion of a spherical body with a displaced center of mass on a surface undergoing oscillations in the vertical plane. The phenomenon of vibrostabilization of an unstable position in which the center of mass of the body is above the geometric center of the sphere is revealed. The influence of the frequency and amplitude of oscillations of the surface on the stabilization of the unstable position of the motion of the spherical body is analyzed.

Mathematics Subject Classification
74H45; 70-05, 70E50

Keywords
spherical top, vibrating plane, stabilization, experiment

DOI
https://doi.org/10.2298/TAM241204001K

Erratum to: Symmetries and stability of motions in the Newtonian and the Hookean potentials (Theor. Appl. Mech. 49(1) (2022), 61--69) DOI: https://doi.org/10.2298/TAM220213005C
Christian Carimalo
Available online 28 February 2025

Abstract

Mathematics Subject Classification

Keywords

DOI
https://doi.org/10.2298/TAM231221002E

On the existence of geodesic vector fields on closed surfaces
Vladimir S. Matveev
Available online 14 April 2025

Abstract
We construct an example of a Riemannian metric on the 2-torus such that its universal cover does not admit global Riemann normal coordinates.

Mathematics Subject Classification
37J30, 53C22, 37J35, 70H06, 53B20

Keywords
geodesic vector fields, Riemann normal coordinates, geodesic normal coordinates, semi-geodesic coordinates, integrable geodesic flow

DOI
https://doi.org/10.2298/TAM241210003M

On cusps of caustics by reflection in two dimensional projective Finsler metrics
Serge Tabachnikov
Available online 15 April 2025

Abstract
Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor.

Mathematics Subject Classification
78A05; 37C83, 53A04

Keywords
caustic, Finsler billiards, projective Finsler metrics

DOI
https://doi.org/10.2298/TAM250109004T

First steps towards the averaging with respect to a part of the coordinates
Ivan Polekhin
Available online 22 April 2025

Abstract
The problem of averaging on an infinite time interval is considered. The classical results on averaging proved by N.N. Bogoluybov are generalized to the case in which only a part of the coordinates in the phase space remains close to the equilibrium position of the averaged system. We call this the averaging with respect to a part of the coordinates. The results are based on some topological ideas combined with the standard theorem on averaging on a finite time interval.

Mathematics Subject Classification
34C29

Keywords
periodic systems, averaging, rapid oscillations, infinite time interval, topological methods in dynamics

DOI
https://doi.org/10.2298/TAM250110005P