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# 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. Keywords: Hamiltonian system, stability, symplectic map, normal form, oscillations, satellite Citation: 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, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 127-144 DOI:10.1134/S1560354719020011
 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 one-dimensional trajectories fixed in the ball’s and disk’s frame. The dynamics of a ball and disk actuated by masses moving along one-dimensional trajectories are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed. Keywords: nonholonomic mechanics, holonomic mechanics, rolling balls, rolling disks Citation: Putkaradze V.,  Rogers S. M., On the Normal Force and Static Friction Acting on a Rolling Ball Actuated by Internal Point Masses, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 145-170 DOI:10.1134/S1560354719020023
 Tsiganov A. V. Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane Abstract We discuss a non-Hamiltonian 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. Keywords: nonholonomic mechanics, separation of variables, Chaplygin ball Citation: Tsiganov A. V., Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 171-186 DOI:10.1134/S1560354719020035
 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. Keywords: Moser path method, volume forms, singularities, $b$-symplectic manifolds Citation: Cardona R.,  Miranda E., On the Volume Elements of a Manifold with Transverse Zeroes, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 187-197 DOI:10.1134/S1560354719020047
 Inozemtsev V. I. On the Structure of Solutions of the Elliptic Calogero – Moser Many-particle Problem Abstract I describe a finite-dimensional manifold which contains all meromorphic solutions to the many-particle 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. Keywords: integrability, Riemann theta functions, three-particle problem Citation: Inozemtsev V. I., On the Structure of Solutions of the Elliptic Calogero – Moser Many-particle Problem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 198-201 DOI:10.1134/S1560354719020059
 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. 152 881-901) proved the existence of an eight-shaped orbit of the planar three-body 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 three-body 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. Keywords: variational methods, symbolic dynamics, periodic solutions Citation: Shibayama M., Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 202-211 DOI:10.1134/S1560354719020060
 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. Keywords: integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber body model, permanent rotations, dynamics in a fixed reference frame, resonance, quadrature, unbounded motion Citation: Kilin A. A.,  Pivovarova E. N., Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 212-233 DOI:10.1134/S1560354719020072

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