Volume 17, Number 1
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.

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.

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 threebody problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh. 
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.

Su T.
On the Accuracy of Conservation of Adiabatic Invariants in SlowFast Hamiltonian Systems
Abstract
Let the adiabatic invariant of action variable in a slowfast 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 slowfast 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.

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 nonautonomous systems have a special resonance zone. The structure of this zone is investigated.

Tsiganov A. V.
One Invariant Measure and Different Poisson Brackets for Two NonHolonomic Systems
Abstract
We discuss the nonholonomic 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 nonzero torsion on configuration space, in contrast with the wellknown Eisenhart–Benenti and Turiel constructions.

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.

Saccomandi G., Vitolo R.
A Translation of the T. LeviCivita paper "Interpretazione Gruppale degli Integrali di un Sistema Canonico" (Rend. Acc. Lincei, 1899, s. 3^{a}, vol.VII, pp. 235–238)
Abstract
In this paper we provide a translation of a paper by T. LeviCivita, 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.
