The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).

We consider the problem of finding integrals of motion for quantum elliptic Calogero–Moser systems with arbitrary number of particles extended by introducing spinexchange interaction. By direct calculation, after making certain ansatz, we found first two integrals — quite probably, lowest nontrivial members of the whole commutative ring. This result might be considered as the first step in constructing this ring of the operators which commute with the Hamiltonian of the model.

The paper is devoted to explicit integration of the classical generalization of the Euler top: the Zhukovski–Volterra system describing the free motion of a gyrostat. We revise the solution for the components of the angular momentum first obtained by Volterra in [1] and present an alternative solution based on an algebraic parametrization of the invariant curves. This also enables us to derive an effective description of the motion of the body in space.

Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddle-focus loops.

We study the behavior of a class of convolution-type nonlinear transformations. Under some smallness conditions we prove the existence of fixed points and analyze the spectrum of the associated linearized operator.

We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k > 2$. The well known Morales–Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set $\mathcal{M}_k \subset \mathbb{Q}$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\bf d)$ calculated at a non-zero $\bf d \in \mathbb{C}^n$ satisfying $V'(\bf d) = \bf d$, belong to $\mathcal{M}_k$.
The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set $\mathcal{I}_{n, k} \subset \mathcal{M}_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\bf d)$ belong to $\mathcal{I}_{n, k}$. We give an algorithm which allows to find sets $\mathcal{I}_{n, k}$.
We applied this results for the case $n = k = 3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.

We prove that an $n$-gon $\mathcal{F}$ is pedal strong Fagnano trajectory in some convex polygonal billiard table $\mathcal{A}$ if and only if it is a convex Poncelet $n$-gon.