Volume 24, Number 2
Volume 24, Number 2, 2019
Bardin B. S., Chekina E. A.
On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Case of Combinational Resonance
Abstract
We deal with a Hamiltonian system with two degrees of freedom,
whose Hamiltonian is a 2$\pi$periodic function of time and analytic
in a neighborhood of an equilibrium point. It is assumed that the
characteristic equation of the system linearized in a neighborhood
of the equilibrium point has two different double roots such that
their absolute values are equal to unity, i.\,e., a combinational
resonance takes place in this system. We consider the case of
general position when the monodromy matrix of the linearized system
is not diagonalizable. In this case the equilibrium point is linearly unstable. However,
this does not imply its instability in the original nonlinear system. Rigorous
conclusions on the stability can be formulated in terms of
coefficients of the Hamiltonian normal form.
We describe a constructive algorithm for constructing and normalizing the symplectic map generated by the phase flow of the Hamiltonian system considered. We obtain explicit relations between the coefficients of the generating function of the symplectic map and the coefficients of the Hamiltonian normal form. It allows us to formulate conditions of stability and instability in terms of coefficients of the above generating function. The developed algorithm is applied to solve the stability problem for oscillations of a satellite with plate mass geometry, that is, $J_z = J_x +J_y$, where $J_x$, $J_y$, $J_z$ are the principal moments of inertia of the satellite, when the parameter values belong to a boundary of linear stability. 
Putkaradze V., Rogers S. M.
On the Normal Force and Static Friction Acting on a Rolling Ball Actuated by Internal Point Masses
Abstract
The goal of this paper is to investigate the normal and tangential forces acting at
the point of contact between a horizontal surface and a rolling ball actuated by internal point
masses moving in the ball’s frame of reference. The normal force and static friction are derived
from the equations of motion for a rolling ball actuated by internal point masses that move
inside the ball’s frame of reference, and, as a special case, a rolling disk actuated by internal
point masses. The masses may move along onedimensional trajectories fixed in the ball’s and
disk’s frame. The dynamics of a ball and disk actuated by masses moving along onedimensional
trajectories are simulated numerically and the minimum coefficients of static friction required
to prevent slippage are computed.

Tsiganov A. V.
Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane
Abstract
We discuss a nonHamiltonian vector field appearing in considering the partial motion of a Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In two partial cases this vector field is expressed via Hamiltonian vector fields using a nonalgebraic deformation of the canonical Poisson bivector on $e^*(3)$. For the symmetric ball we also calculate variables of separation, compatible Poisson brackets, the algebra of Haantjes operators and $2\times2$ Lax matrices.

Cardona R., Miranda E.
On the Volume Elements of a Manifold with Transverse Zeroes
Abstract
Moser proved in 1965 in his seminal paper [15] that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from
a hypersurface where they fulfill a transversality assumption ($b$Poisson structures). We do this using the desingularization technique introduced in [10] and extend it to $b^m$Nambu structures. 
Inozemtsev V. I.
On the Structure of Solutions of the Elliptic Calogero – Moser Manyparticle Problem
Abstract
I describe a finitedimensional manifold which contains all meromorphic solutions to the manyparticle elliptic Calogero – Moser problem at some fixed values of the coupling constant. These solutions can be selected by purely algebraic calculations as it was shown in the simplest case of three interacting particles.

Shibayama M.
Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem
Abstract
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881901) proved the existence of an eightshaped orbit of the planar threebody problem with equal masses. Since then a number of solutions to the $N$body problem have been discovered. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the threebody problem. The system is known to be chaotic and was studied by using symbolic dynamics (J.Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973).
In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits. 
Kilin A. A., Pivovarova E. N.
Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges
Abstract
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the
plane of sections are presented.
