The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.

Vu Dong Tô has proven in [1] that for any mapping $f: X \to X$, where $X$ is a metric space that is not precompact, the third condition in the Devaney’s definition of chaos follows from the first two even if $f$ is not assumed to be continuous. This paper completes this result by analysing the precompact case. We show that if $X$ is either finite or perfect one can always find a map $f: X \to X$ that satisfies the first two conditions of Devaney’s chaos but not the third. Additionally, if $X$ is neither finite nor perfect there is no $f: X \to X$ that would satisfy the first two conditions of Devaney’s chaos at the same time.

Using the technique of asymptotic expansions, we calculate trajectories of three point vortices in the vicinity of stable equilateral or collinear configurations. We show that in an appropriate rotating coordinate system each vortex moves in an elliptic orbit. The orbits of the vortices have equal eccentricities. The angle and ratio between the major axes of any two orbits have a simple analytic representation.

Basic investigation techniques, algorithms, and results are presented for nonlinear oscillations and stability of steady rotations and periodic motions of a rigid body, colliding with a rigid surface, in a uniform gravity field.

Following closely Kolmogorov’s original paper [1], we give a complete proof of his celebrated Theorem on perturbations of integrable Hamiltonian systems by including few "straightforward" estimates.