A variant of Kolmogorov’s initial proof is given, in terms of a group of symplectic transformations and of an elementary fixed point theorem.

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.

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.

This paper studies the topology of the constant energy surfaces of the double spherical pendulum.

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.

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.

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.

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.

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.