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Vol. 1, No. 2

Vol. 1, No. 2, 2005

Grigoryev Y. A.,  Tsiganov A. V.
We discuss an algorithm for calculating of the separated variables for the Hamilton-Jacobi equation for the wide class of the so-called L-systems on the Riemann manifolds of the constant curvature. We suggest a software implementation of this algorithm in the system of symbolic computations Maple and consider several examples.
Keywords: integrable systems, Hamilton-Jacobi equation, separation of variables
Citation: Grigoryev Y. A.,  Tsiganov A. V., Computing of the separated variables for the Hamilton-Jacobi equation on a computer, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 163-179
Kholostova O. V.
We study the motion of a satellite (a rigid body) in a circular orbit about its centre of mass. The satellite is subject to the central Newtonian gravitational field. The satellite’s principal central moments of inertia $A$, $B$ and $C$ are assumed to satisfy the equation $B=A+C$. This equation holds for thin plates. Particular motions occur when the plate executes pendulum-like oscillations of an arbitrary amplitude in the plane of the orbit. A linear analysis of the orbital stability of this motion is carried out. In the plane of parameters of the problem (an amplitude of oscillations and an inertial parameter) domains of orbital linear stability and instability of oscillations of the satellite are obtained both numerically and analytically.
Keywords: satellite, orbital stability, parametric resonance, action-angle variables, Deprit-Hori method
Citation: Kholostova O. V., Linear analysis of stability the planar oscillations of a satellite being a plate in a circular orbit, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 181-190
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
The paper deals with a transition to chaos in the phase-plane portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth of the homoclinic structure and 2) development of cascades of period doubling bifurcations. On the zero level of the integral of areas, an adiabatic behavior of the system (as the energy tends to zero) has been noticed. Meander tori induced by the breakdown of the torsion property of the mapping have been found.
Keywords: motion of a rigid body, phase-plane portrait, mechanism of chaotisation, bifurcations
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Chaos in a restricted problem of rotation of a rigid body with a fixed point, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 191-207
Tronin K. G.
The paper explores the evolution of rotation of a rigid body influenced by a constant and dissipative disturbing moments. With the assumption that the disturbing moments are small, it has been shown numerically that for almost all initial conditions the body’s motion tends asymptotically to a steady rotation around a principal axis with either largest or smallest moment of inertia. On the plane of initial conditions, the points corresponding to these two types of ultimate rotation have been shown to be distributed almost randomly.
Keywords: disturbed motion, probabilistic phenomena, diagrams of asymptotic motion
Citation: Tronin K. G., Numerical analysis of rotation of a rigid body subject to the sum of a constant and dissipative moment, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 209-213
Galperin G. A.
The paper concerns the behavior of four gravitating bodies. Three bodies have zero mass and are referred to «planet». The fourth body (the «Sun») is a massive gravitating center. The conditions under which the four bodies lie in one plane are discussed. These conditions are shown to be deeply connected with arithmetic and geometric characteristics of «planets» motion.
Keywords: flattening moment, gravitating bodies, motion of planets, torus
Citation: Galperin G. A., Two-dimensional configuration in the three-dimensional problem of four gravitating bodies, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 215-233
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords: reduction, point vortex, equations of motion, Poincare map
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 233-246
Kessler P.,  O'Reilly O. M.
The motion of disks spun on tables has the well-known feature that the associated acoustic signal increases in frequency as the motion tends towards its abrupt halt. Recently, a commercial toy, known as Euler’s disk, was designed to maximize the time before this abrupt ending. In this paper, we present and simulate a rigid body model for Euler’s disk. Based on the nature of the contact force between the disk and the table revealed by the simulations, we conjecture a new mechanism for the abrupt halt of the disk and the increased acoustic frequency associated with the decline of the disk.
Keywords: rigid body, motion of disk, equations of motion, dissipation
Citation: Kessler P.,  O'Reilly O. M., The ringing of Euler's disk, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 247-260
Diacu F.
Citation: Diacu F., Singularuties of the Newtonian N-body problem, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 261-280

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