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

Unusual properties of adiabatic invariance in a billiard model related to the adiabatic Piston problem
Joshua Skinner and Anatoly Neishtadt
Available online 05 May 2025

Abstract
We consider the motion of two massive particles along a straight line. A lighter particle bounces back and forth between a heavier particle and a stationary wall, with all collisions being ideally elastic. This is one of canonical models in the theory of adiabatic invariants. It is known that if the lighter particle moves much faster than the heavier one, and the kinetic energies of the particles are of the same order, then the product of the speed of the lighter particle and the distance between the heavier particle and the wall is an adiabatic invariant: its value remains approximately constant over a long period. We show that the value of this adiabatic invariant, calculated at the collisions of the lighter particle with the wall, is a constant of motion (i.e., \emph{an exact adiabatic invariant}). On the other hand, the value of this adiabatic invariant at the collisions between the particles slowly, linearly in time, decays with each collision.
The model we consider is a highly simplified version of the classical adiabatic piston problem, where the lighter particle represents a gas particle, and the heavier particle represents the piston.

Mathematics Subject Classification
70H11

Keywords
adiabatic invariant, elastic collisions

DOI
https://doi.org/10.2298/TAM241121006S

Nipping analysis of a two-leaf spring strengthened by an additional full-length leaf
Vlado A. Lubarda, Marko V. Lubarda
Available online 15 May 2025

Abstract
Nipping analysis of a rectangular two-leaf spring strengthened by an additional full-length leaf is presented. Closed form expressions are derived for the initial gaps between the leaves which make the maximum stresses in all the leaves within the clamped cross-section of the loaded spring equal to each other. The derived expressions are general in the sense that they apply for any values of the introduced leaf-length and thickness parameters. Two initial gaps between the pairs of consecutive leaves are needed to achieve the desired stress reduction. The required gaps can be either positive or negative, depending on the values of introduced spring parameters. For some combination of length and thickness parameters, nipping is not an effective means of stress reduction. There is a particular combination of parameters for which the maximum stresses in all critical cross-sections of leaves become equal to each other. The presented analysis and obtained results may be useful for multi-leaf spring design and related optimization studies.

Mathematics Subject Classification
74K10, 74B05

Keywords
contact force, curvature, gap, leaf spring, maximum stress, nipping, spring constant

DOI
https://doi.org/10.2298/TAM241225007L

On non-approximability of zero loss global ℒ² minimizers by gradient descent in deep learning
Thomas Chen and Patricia Muñoz Ewald
Available online 20 May 2025

Abstract
We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL), and give a detailed discussion of the circumstance that, in underparametrized DL networks, zero loss minimization cannot generically be attained. As a consequence, we conclude that the distribution of training inputs must necessarily be non-generic in order to produce zero loss minimizers, both for the method constructed in [2.3], or for gradient descent [1] (which assume clustering of training data).

Mathematics Subject Classification
57R70, 62M45

Keywords
deep learning, underparametrization, generic training data, zero loss

DOI
https://doi.org/10.2298/TAM250121008C