Volume 28, Number 6
Volume 28, Number 6, 2023
Tsiganov A. V.
Abstract
Affine transformations in Euclidean space generate a correspondence between
integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in
$R^n$. Using this correspondence and the suitable coupling constant transformations, we can get
real integrals of motion in the hyperboloid case starting with real integrals of motion in the
sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic
and sextic polynomials in momenta.

Arathoon P.
Abstract
The $2$body problem on the sphere and hyperbolic space are both real forms
of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural
description in terms of biquaternions and allows us to address questions concerning the
hyperbolic system by complexifying it and treating it as the complexification of a spherical
system. In this way, results for the $2$body problem on the sphere are readily translated to
the hyperbolic case. For instance, we implement this idea to completely classify the relative
equilibria for the $2$body problem on hyperbolic $3$space and discuss their stability for a strictly
attractive potential.

BravoDoddoli A.
Abstract
The space of $2$jets of a real function of two real variables, denoted by $J^2(\mathbb{R}^2,\mathbb{R})$, admits the structure of a metabelian Carnot group, so $J^2(\mathbb{R}^2,\mathbb{R})$ has a normal abelian subgroup $\mathbb{A}$. As any subRiemannian manifold, $J^2(\mathbb{R}^2,\mathbb{R})$ has an associated Hamiltonian geodesic flow. The Hamiltonian action of $\mathbb{A}$ on $T^*J^2(\mathbb{R}^2,\mathbb{R})$ yields the reduced Hamiltonian $H_{\mu}$ on $T^*\mathcal{H} \simeq T^*(J^2(\mathbb{R}^2,\mathbb{R})/\mathbb{A})$, where $H_{\mu}$ is a twodimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian $H_{\mu}$ is nonintegrable by meromorphic functions for some values of $\mu$. This result suggests the subRiemannian geodesic flow on $J^{2}(\mathbb{R}^2,\mathbb{R})$ is not meromorphically integrable.

Pinzari G.
Abstract
We review a recent application of the ideas of normal form theory to systems
(Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate
variable. The main difference from the standard case consists in the nonuniqueness of the
normal form and the total absence of the small divisors problem. The exposition is quite general,
so as to allow extensions to the case of more nonperiodic coordinates, and more functional
settings. Here, for simplicity, we work in the realanalytic class.

Galkin V. D., Nozdrinova E. V., Pochinka O. V.
Abstract
In this paper, we obtain a classification of gradientlike flows on arbitrary surfaces
by generalizing the circular Fleitas scheme. In 1975 he proved that such a scheme is a complete
invariant of topological equivalence for polar flows on 2 and 3manifolds. In this paper, we
generalize the concept of a circular scheme to arbitrary gradientlike flows on surfaces.We prove
that the isomorphism class of such schemes is a complete invariant of topological equivalence.
We also solve exhaustively the realization problem by describing an abstract circular scheme
and the process of realizing a gradientlike flow on the surface. In addition, we construct an
efficient algorithm for distinguishing the isomorphism of circular schemes.

Bardin B. S.
Abstract
A general method is presented for constructing a nonlinear canonical transformation,
which makes it possible to introduce local variables in a neighborhood of periodic motions
of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for
investigating the behavior of the Hamiltonian system in the vicinity of its periodic trajectories.
In particular, it can be applied to solve the problem of orbital stability of periodic motions.

Kilin A. A., Ivanova T. B., Pivovarova E. N.
Abstract
This paper treats the problem of a spherical robot with an axisymmetric pendulum
drive rolling without slipping on a vibrating plane. The main purpose of the paper is to
investigate the stabilization of the upper vertical rotations of the pendulum using feedback
(additional control action). For the chosen type of feedback, regions of asymptotic stability
of the upper vertical rotations of the pendulum are constructed and possible bifurcations are
analyzed. Special attention is also given to the question of the stability of periodic solutions
arising as the vertical rotations lose stability.
