We present a new derivation of the local structure of Koenigs metrics using a framework laid down by Matveev and Shevchishin. All of these dynamical systems allow for a potential preserving their superintegrability (SI) and most of them are shown to be globally defined on either ${\mathbb R}^2$ or ${\mathbb H}^2$. Their geodesic flows are easily determined thanks to their quadratic integrals. Using Carter (or minimal) quantization, we show that the formal SI is preserved at the quantum level and for two metrics, for which all of the geodesics are closed, it is even possible to compute the classical action variables and the point spectrum of the quantum Hamiltonian.

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential
$U(q, t) = f(t)V (q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to\pm\infty$ and vanishes at a unique point $t_{0} \in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x, t)$ as a function of $x$ distinguishes its maximum for any fixed $t > t_{0}$ and $t < t_{0}$, respectively. Under nondegeneracy conditions on points of $X_\pm$ we prove the existence of infinitely many doubly asymptotic trajectories connecting $X_{-}$ and $X_{+}$.

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper [14] for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, Pérez-Chavela and Reyes Victoria [9], we apply stereographic projection and study the problem in the Poincaré disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in [14]. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.

We review our recent results concerning the propagation of “quasi-particles” in hybrid spaces — topological spaces obtained from graphs via replacing their vertices by Riemannian manifolds. Although the problem is purely classical, it is initiated by the quantum one, namely, by the Cauchy problem for the time-dependent Schrödinger equation with localized initial data.We describe connections between the behavior of quasi-particles with the properties of the corresponding geodesic flows. We also describe connections of our problem with various problems in analytic number theory.

We consider the problem of the inclusion of a diffeomorphism into a flow generated by an autonomous or time periodic vector field. We discuss various aspects of the problem, present a series of results (both known and new ones) and point out some unsolved problems.

We study a family of Liénard-type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as travelingwave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this family of equations. To this end we use an approach which is based on the application of nonlocal transformations. By studying connections between this family of Liénard-type equations and type III Painlevé–Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Liénard-type equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Liénard-type equations and other integrability conditions for this family of equations.

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.

The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $L-A$ pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler – van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler – van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on $so(4)$.