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# Volume 17, Number 1, 2012

 Féjoz J. A Proof of the Invariant Torus Theorem of Kolmogorov Abstract A variant of Kolmogorov’s initial proof is given, in terms of a group of symplectic transformations and of an elementary fixed point theorem. Keywords: Celestial Mechanics Citation: Féjoz J., A Proof of the Invariant Torus Theorem of Kolmogorov, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 1-5 DOI:10.1134/S1560354712010017
 Marchal C. Progress in Celestial Mechanics Abstract Celestial Mechanics has a long history strongly related to human condition and to the successive scientific revolutions. It remains the most accurate science and has seen tremendous progress in the three last centuries. It has led to the discovery of several dissipative phenomena that are in many cases the real factors of evolution. The unsolved questions of Celestial Mechanics may have a decisive impact on cosmology. Keywords: Celestial Mechanics, synthesis, cosmology Citation: Marchal C., Progress in Celestial Mechanics, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 6-23 DOI:10.1134/S1560354712010029
 Meyer K. R.,  Palacián J. F.,  Yanguas P. Stability of a Hamiltonian System in a Limiting Case Abstract We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the $1 : −1$ resonance case where the linearized system has double pure imaginary eigenvalues $\pm i \omega$, $\omega \ne 0$ and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable. This result implies the stability of the Lagrange equilateral triangle libration points, $\mathcal{L}_4$ and $\mathcal{L}_5$, in the planar circular restricted three-body problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh. Keywords: stability, Lagrange equilateral triangle, KAM tori, Liapunov stable, planar circular restricted three-body problems, Routh’s critical mass ratio Citation: Meyer K. R.,  Palacián J. F.,  Yanguas P., Stability of a Hamiltonian System in a Limiting Case, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 24-35 DOI:10.1134/S1560354712010030
 Hu S.,  Leandro E.,  Santoprete M. On the Topology of the Double Spherical Pendulum Abstract This paper studies the topology of the constant energy surfaces of the double spherical pendulum. Keywords: double spherical pendulum, topology of level sets, Morse theory, Gysin sequence Citation: Hu S.,  Leandro E.,  Santoprete M., On the Topology of the Double Spherical Pendulum, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 36-53 DOI:10.1134/S1560354712010042
 Su T. On the Accuracy of Conservation of Adiabatic Invariants in Slow-Fast Hamiltonian Systems Abstract Let the adiabatic invariant of action variable in a slow-fast Hamiltonian system with two degrees of freedom have limits along the trajectories as time tends to plus and minus infinity. The difference of these two limits is exponentially small in analytic systems. An isoenergetic reduction and canonical transformations are applied to transform the slow-fast system to form of a system depending on a slowly varying parameter in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given. Keywords: adiabatic invariant, slow-fast Hamiltonian systems, isoenergetic reduction Citation: Su T., On the Accuracy of Conservation of Adiabatic Invariants in Slow-Fast Hamiltonian Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 54-62 DOI:10.1134/S1560354712010054
 Morozov A. D.,  Mamedov E. A. On a Double Cycle and Resonances Abstract We consider systems with 3/2 degrees of freedom close to nonlinear autonomous Hamiltonian ones in the case where the perturbed autonomous systems have a double limit cycle. Then the initial non-autonomous systems have a special resonance zone. The structure of this zone is investigated. Keywords: cycle, resonances, average Citation: Morozov A. D.,  Mamedov E. A., On a Double Cycle and Resonances, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 63-71 DOI:10.1134/S1560354712010066
 Tsiganov A. V. One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems Abstract We discuss the non-holonomic Chaplygin and the Borisov–Mamaev–Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by $L$-tensors with non-zero torsion on configuration space, in contrast with the well-known Eisenhart–Benenti and Turiel constructions. Keywords: non-holonomic mechanics, Chaplygin’s rolling ball, Poisson brackets Citation: Tsiganov A. V., One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 72-96 DOI:10.1134/S1560354712010078
 Ivanov A. P.,  Shuvalov N. D. On the Motion of a Heavy Body with a Circular Base on a Horizontal Plane and Riddles of Curling Abstract In this paper we discuss the dynamics of an axisymmetric rigid body whose circular area moves upon a horizontal rough surface. We investigate the interaction between the character of the law of friction and the curvature of the body’s trajectory. For the case of a curling stone it is shown that the observed effects can only be explained using the dependence of the friction coefficient on the Gümbel number. The procedure for constructing the law of friction based on experimental data is developed. It is shown that the available data can only be substantiated by means of anisotropic friction. The simplest model of such friction is constructed which provides quantitative coincidence with the experiment. Keywords: mixed friction, Stribeck’s curve, dynamics of curling stones Citation: Ivanov A. P.,  Shuvalov N. D., On the Motion of a Heavy Body with a Circular Base on a Horizontal Plane and Riddles of Curling, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 97-104 DOI:10.1134/S156035471201008X
 Saccomandi G.,  Vitolo R. A Translation of the T. Levi-Civita paper "Interpretazione Gruppale degli Integrali di un Sistema Canonico" (Rend. Acc. Lincei, 1899, s. 3a, vol.VII, pp. 235–238) Abstract In this paper we provide a translation of a paper by T. Levi-Civita, published in 1899, about the correspondence between symmetries and conservation laws for Hamilton’s equations. We discuss the results of this paper and their relationship with the more general classical results by E. Noether. Keywords: Levi-Civita, point symmetries, conservation laws, Hamilton’s equations, Noether’s theorem Citation: Saccomandi G.,  Vitolo R., A Translation of the T. Levi-Civita paper "Interpretazione Gruppale degli Integrali di un Sistema Canonico" (Rend. Acc. Lincei, 1899, s. 3a, vol.VII, pp. 235–238), Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 105-112 DOI:10.1134/S1560354712010091

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