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# Volume 22, Number 4, 2017

 Valent G. Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I) Abstract We present a family of superintegrable (SI) systems which live on a Riemannian surface of revolution and which exhibit one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2, one recovers a metric due to Koenigs. The local structure of these systems is under control of a linear ordinary differential equation of order $n$ which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called “simple” case (see Definition 2). Some globally defined examples are worked out which live either in $\mathbb{H}^2$ or in $\mathbb{R}^2$. Keywords: superintegrable two-dimensional systems, differential systems, ordinary differential equations, analysis on manifolds Citation: Valent G., Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I), Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 319-352 DOI:10.1134/S1560354717040013
 Tsiganov A. V. Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball Abstract The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order. Keywords: nonholonomic systems, Abel quadratures, arithmetic of divisors Citation: Tsiganov A. V., Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 353-367 DOI:10.1134/S1560354717040025
 Rauch-Wojciechowski S.,  Przybylska M. Understanding Reversals of a Rattleback Abstract A counterintuitive unidirectional (say counterclockwise) motion of a toy rattleback takes place when it is started by tapping it at a long side or by spinning it slowly in the clockwise sense of rotation. We study the motion of a toy rattleback having an ellipsoidal-shaped bottom by using frictionless Newton equations of motion of a rigid body rolling without sliding in a plane. We simulate these equations for tapping and spinning initial conditions to see the contact trajectory, the force arm and the reaction force responsible for torque turning the rattleback in the counterclockwise sense of rotation. Long time behavior of such a rattleback is, however, quasi-periodic and a rattleback starting with small transversal oscillations turns in the clockwise direction. Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, numerical solutions Citation: Rauch-Wojciechowski S.,  Przybylska M., Understanding Reversals of a Rattleback, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 368-385 DOI:10.1134/S1560354717040037
 Combot T. Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus Abstract We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong rational integrability condition on $V$, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimension 2 and 3, and recover several integrable cases. These potentials after a complex variable change become real, and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high degree first integrals are explicitly integrated. Keywords: trigonometric polynomials, differential Galois theory, integrability, Toda lattice Citation: Combot T., Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 386-407 DOI:10.1134/S1560354717040049
 Cieliebak K.,  Frauenfelder U.,  van Koert O. Periodic Orbits in the Restricted Three-body Problem and Arnold’s $J^+$ -invariant Abstract We apply Arnold’s theory of generic smooth plane curves to Stark – Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold’s $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark – Zeeman systems and study their behavior. Keywords: generic immersions into the plane, Arnold’s plane curve invariants, restricted threebody problem Citation: Cieliebak K.,  Frauenfelder U.,  van Koert O., Periodic Orbits in the Restricted Three-body Problem and Arnold’s $J^+$ -invariant, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 408-434 DOI:10.1134/S1560354717040050
 Borisov A. V.,  Mamaev I. S. An Inhomogeneous Chaplygin Sleigh Abstract In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics. Keywords: Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance Citation: Borisov A. V.,  Mamaev I. S., An Inhomogeneous Chaplygin Sleigh, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 435-447 DOI:10.1134/S1560354717040062
 Knauf A.,  Seri M. Symbolic Dynamics of Magnetic Bumps Abstract For $n$ convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one. Keywords: magnetic billiards, symbolic dynamics, classical mechanics Citation: Knauf A.,  Seri M., Symbolic Dynamics of Magnetic Bumps, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 448-454 DOI:10.1134/S1560354717040074

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