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# Volume 17, Number 6, 2012

 Labrousse C. Flat Metrics are Strict Local Minimizers for the Polynomial Entropy Abstract As we have proved in [11], the geodesic flows associated with the flat metrics on $\mathbb{T}^2$ minimize the polynomial entropy $h_{pol}$. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated with flat metrics are local strict minima for $h_{pol}$. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems. Keywords: geodesic flows, polynomial entropy, integrable systems Citation: Labrousse C., Flat Metrics are Strict Local Minimizers for the Polynomial Entropy, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 479-491 DOI:10.1134/S1560354712060019
 Dobrogowska A.,  Odzijewicz A. Integrable Relativistic Systems Given by Hamiltonians with Momentum-spin-orbit Coupling Abstract In the paper we investigate evolution of the relativistic particle (massive and massless) with spin defined by Hamiltonian containing the terms with momentum-spin-orbit coupling. We integrate the corresponding Hamiltonian equations in quadratures and express their solutions in terms of elliptic functions. Keywords: bi-Hamiltonian, integrable systems, relativistic particle, twistor space Citation: Dobrogowska A.,  Odzijewicz A., Integrable Relativistic Systems Given by Hamiltonians with Momentum-spin-orbit Coupling, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 492-505 DOI:10.1134/S1560354712060020
 Pivovarov M. On the Probability of the Outcomes in Buckling of an Elastic Beam Abstract A nonlinear time-varying one-degree-of-freedom system, which is used for the modelling of the buckling of a loaded beam in Euler’s problem, is considered. For a slowly changing load, the deterministic approach in this problem fails if the trajectories pass through the separatrix. An expression for the probability of possible outcomes of the evolution of the oscillations is obtained. The analytical and numerical results are compared. Keywords: deterministic chaos, separatrix crossing, buckling of beams Citation: Pivovarov M., On the Probability of the Outcomes in Buckling of an Elastic Beam, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 506-511 DOI:10.1134/S1560354712060032
 Borisov A. V.,  Jalnine A. Y.,  Kuznetsov S. P.,  Sataev I. R.,  Sedova Y. V. Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback Abstract We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented. Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension Citation: Borisov A. V.,  Jalnine A. Y.,  Kuznetsov S. P.,  Sataev I. R.,  Sedova Y. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532 DOI:10.1134/S1560354712060044
 Bardin B. S.,  Rudenko T. V.,  Savin A. A. On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev–Steklov Case Abstract We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev–Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, orbital stability Citation: Bardin B. S.,  Rudenko T. V.,  Savin A. A., On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev–Steklov Case, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 533-546 DOI:10.1134/S1560354712060056
 Ramodanov S. M.,  Tenenev V. A.,  Treschev D. V. Self-propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid Abstract We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms. Keywords: perfect fluid, self-propulsion, Flettner rotor Citation: Ramodanov S. M.,  Tenenev V. A.,  Treschev D. V., Self-propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 547-558 DOI:10.1134/S1560354712060068
 Bolotin S. V. The Problem of Optimal Control of a Chaplygin Ball by Internal Rotors Abstract We study the problem of optimal control of a Chaplygin ball on a plane by means of 3 internal rotors. Using Pontryagin maximum principle, the equations of extremals are reduced to Hamiltonian equations in group variables. For a spherically symmetric ball, the solutions can be expressed in by elliptic functions. Keywords: nonholonomic constraint, vaconomic mechanics, optimal control, maximum principle, Hamiltonian Citation: Bolotin S. V., The Problem of Optimal Control of a Chaplygin Ball by Internal Rotors, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 559-570 DOI:10.1134/S156035471206007X
 Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S. Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals Abstract In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles. Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 571-579 DOI:10.1134/S1560354712060081
 Kozlov V. V. An Extended Hamilton–Jacobi Method Abstract We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method. Keywords: generalized Lamb’s equations, vortex manifolds, Clebsch potentials, Lagrange brackets Citation: Kozlov V. V., An Extended Hamilton–Jacobi Method, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 580-596 DOI:10.1134/S1560354712060093
 Marchal C. Addendum to "Progress in Celestial Mechanics" [RCD, 2012, vol. 17, no. 1, pp. 6–23] Abstract We use this addendum to indicate a remarkable and simple generalization of a classical property of two-body motions. Keywords: Celestial Mechanics, synthesis, cosmology Citation: Marchal C., Addendum to "Progress in Celestial Mechanics" [RCD, 2012, vol. 17, no. 1, pp. 6–23], Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 597-598 DOI:10.1134/S156035471206010X

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