Volume 24, Number 5
Volume 24, Number 5, 2019
Sergey Chaplygin Memorial Issue
GarcíaNaranjo L. C.
Integrability of the $n$dimensional Axially Symmetric Chaplygin Sphere
Abstract
We consider the $n$dimensional Chaplygin sphere under the assumption that the mass distribution of the sphere is axisymmetric. We prove that, for initial conditions whose angular momentum about the contact point is vertical, the dynamics is quasiperiodic. For $n=4$ we perform the reduction by the associated $\mathrm{SO}(3)$ symmetry and show that the reduced system is integrable by the Euler–Jacobi theorem.

Adabrah A. K., Dragović V., Radnović M.
Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials
Abstract
We derive necessary and sufficient conditions for periodic and for elliptic periodic
trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining
elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with
small periods. We observe a relationship between Cayleytype conditions and discriminantly
separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic
periodicity are derived in terms of polynomialfunctional equations as well. The corresponding
polynomials are related to the classical extremal polynomials. In particular, the lightlike
periodic trajectories are related to the classical Chebyshev polynomials. Similarities and
differences with respect to the previously studied Euclidean case are highlighted.

Markeev A. P.
On the Stability of the Regular Precession of an Asymmetric Gyroscope at a Secondorder Resonance
Abstract
The motion of a rigid body about a fixed point in a homogeneous gravitational
field is investigated. The body is not dynamically symmetric and its center of gravity lies on
the perpendicular, raised from the fixed point, to one of the circular sections of an ellipsoid
of inertia. A body with such mass geometry may precess regularly about a nonvertical axis
(Grioli’s precession). The problem of the orbital stability of this precession is solved for critical
cases of secondorder resonance, when terms higher than degree four in the series expansion of
the Hamiltonian of the perturbed motion should be taken into account.

Bustamante M. D., Lynch P.
Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball
Abstract
The dynamics of a spherical body with a nonuniform mass distribution rolling
on a plane were discussed by Sergey Chaplygin, whose 150th birthday we celebrate this year.
The Chaplygin top is a nonintegrable system, with a colourful range of interesting motions.
A special case of this system was studied by Edward Routh, who showed that it is integrable.
The Routh sphere has a centre of mass offset from the geometric centre, but it has an axis of
symmetry through both these points, and equal moments of inertia about all axes orthogonal
to the symmetry axis. There are three constants of motion: the total energy and two quantities
involving the angular momenta.
It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether’s invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation. As a further demonstration of the power and utility of this method, we find the Noetherian symmetries and corresponding integrals for a system introduced recently, the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether’s theorem. 
Esen O., Jiménez V. M., de León M., Sardón C.
Reduction of a Hamilton – Jacobi Equation for Nonholonomic Systems
Abstract
We discuss, in all generality, the reduction of the Hamilton – Jacobi equation for
systems subject to nonholonomic constraints and invariant under the action of a group of
symmetries.We consider nonholonomic systems subject to both linear and nonlinear constraints
and with different positioning of such constraints with respect to the symmetries.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem
Abstract
This paper addresses the problem of the rolling of a spherical shell with a frame
rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire
system is at the geometric center of the shell.
For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists a noninertial coordinate system (attached to the frame) in which the equations of motion do not depend explicitly on time. The resulting equations of motion preserve an analog of the angular momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the problem reduces to investigating a twodimensional Poincaré map. The case of the rubber rolling model is analyzed in detail. Numerical investigation of its Poincaré map shows the existence of chaotic trajectories, including those associated with a strange attractor. In addition, an analysis is made of the case of motion from rest, in which the problem reduces to investigating the vector field on the sphere $S^2$. 