Volume 28, Number 1
Volume 28, Number 1, 2023
Proceeding of GDIS 2022
Bolotin S. V., Treschev D. V.
Abstract
We consider Hamiltonian systems possessing families of nonresonant invariant tori
whose frequencies are all collinear. Then under certain conditions the frequencies depend
on energy only. This is a generalization of the wellknown Gordon’s theorem about periodic
solutions of Hamiltonian systems. While the proof of Gordon’s theorem uses Hamilton’s
principle, our result is based on Percival’s variational principle. This work was motivated by
the problem of isochronicity in Hamiltonian systems.

Dragović V., Radnović M.
Abstract
We study billiard systems within an ellipsoid in the 4dimensional pseudoEuclidean
spaces. We provide an analysis and description of periodic and weak periodic trajectories in
algebrogeometric and functionalpolynomial terms.

Jovanović B., Šukilović T., Vukmirović S.
Abstract
In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra $\mathfrak{g}_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak{g}$ related to the filtrations of Lie algebras
$\mathfrak{g}_0\subset\mathfrak{g}_1\subset\mathfrak{g}_2\dots\subset\mathfrak{g}_{n1}\subset \mathfrak{g}_n=\mathfrak{g}$ are integrable as well.
In particular, by taking $\mathfrak{g}_0=\{0\}$ and natural filtrations of ${\mathfrak{so}}(n)$ and $\mathfrak{u}(n)$, we have
Gel’fand – Cetlin integrable systems. We prove the conjecture
for filtrations of compact Lie algebras $\mathfrak{g}$: the system is integrable in a noncommutative sense by means of polynomial integrals.
Various constructions of complete commutative polynomial integrals for the system are also given.

Dragović V., Gajić B., Jovanović B.
Abstract
We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$.
In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without
slipping in contact with the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations, three of which are new.
We derive the equations of motion and find an invariant measure for these systems.
As the main result, for $n=1$ we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem.
The obtained integrable nonholonomic models are natural extensions of the wellknown Chaplygin ball integrable problems.
Further, we explicitly integrate
the planar problem consisting of $n$ homogeneous balls of the same radius, but with different
masses, which roll without slipping
over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.

Kilin A. A., Pivovarova E. N.
Abstract
The problem of the rolling of a disk on a plane is considered under the assumption
that there is no slipping in the direction parallel to the horizontal diameter of the disk and
that the center of mass does not move in the horizontal direction. This problem is reduced to
investigating a system of three firstorder differential equations. It is shown that the reduced
system is reversible relative to involution of codimension one and admits a twoparameter family
of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation,
the nonintegrability of the problem is shown. It is proved that the reduced system admits, even
in the nonintegrable case, a twoparameter family of periodic solutions. A number of dynamical
effects due to the existence of involution of codimension one and to the degeneracy of the fixed
points of the reduced system are found.

Bizyaev I. A., Mamaev I. S.
Abstract
In this paper we investigate a nonholonomic system with parametric excitation,
a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the
existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration).
We find a criterion for the existence of trajectories for which one of the velocity components
increases withound bound and has asymptotics $t^{1/3}$. In addition, we show that the problem
under consideration reduces to analysis of a threedimensional Poincaré map. This map exhibits
both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.
