Volume 30, Number 4
Special Issue: Celebrating the 75th Birthday of V.V. Kozlov (Issue Editors: Sergey Bolotin, Vladimir Dragović, and Dmitry Treschev)

We show that the equations of motion of a rigid body about a fixed point in the Newtonian field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schr¨odinger operators are discussed.

We explicitly compute the maximal Lyapunov exponent for a switched system on $\mathrm{SL}_2(\mathbb R)$ and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.

In this paper the two-dimensional Lorentzian problem on the anti-de Sitter plane is studied. Using methods of geometric control theory and differential geometry, we describe the reachable set, investigate the existence of Lorentzian length maximizers, compute extremal trajectories, construct an optimal synthesis, characterize Lorentzian distance and spheres, and describe the Lie algebra of Killing vector fields.

In this note, we briefly discuss how the singular KAM theory of [7] — which was worked out for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ — can be extended to convex real-analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by $h(y)+\varepsilon f(x)$ with $h$ convex and $f$ generic.

Nonholonomic systems are mechanical systems with ideal velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper reviews infinite-dimensional and field-theoretic nonholonomic systems as well as Hamel’s formalism for these settings.

The aim of this note is to present a simple observation that a slight refinement of the contraction mapping principle allows one to recover the precise convergence rate in the Picard – Lindelöf theorem.

We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb{R}^n$ and also motion restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $\mathbb{R}^n$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly nontrivial. We prove complete integrability of the obtained restricted magnetic systems for $n\leqslant 6$. The first integrals of motion of the magnetic flows on the spheres $S^{n-1}$, for $n=5$ and $n=6$, are polynomials of degree $1$, $2$, and $3$ in momenta. We prove noncommutative integrability of the obtained magnetic flows for any $n\geqslant 7$ when the systems allow a reduction to the cases with $n\leqslant 6$. We conjecture that the restricted magnetic systems on $S^{n-1}$ are integrable for all $n$.

The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane — affine or projective, when that is not the case, i. e., the conics have at least one point of tangency or at least one of the conics is not smooth. In all such cases, we find necessary and sufficient conditions for the existence of an $n$-gon inscribed in one of the conics and circumscribed about the other.

We describe the Ozawa solution to the Davey – Stewartson II equation from the point of view of surface theory by presenting a soliton deformation of surfaces which is ruled by the Ozawa solution. The Ozawa solution blows up at a certain moment and we describe explicitly the corresponding singularity of the deformed surface.

In [7] an open set of structurally unstable families of vector fields on a sphere was constructed. More precisely, a vector field with a degeneracy of codimension three was discovered whose bifurcation in a generic three-parameter family has a numeric invariant. This vector field has a polycycle and two saddles, one inside and one outside this polycycle; one separatrix of the outside saddle winds towards the polycycle and one separatrix of the inside saddle winds from it. Families with functional invariants were constructed also. In [2] a hyperbolic polycycle with five vertices and no saddles outside it was constructed whose bifurcations in its arbitrary narrow neighborhood (semilocal bifurcations in other words) have a numeric invariant and thus are structurally unstable. This paper deals with semilocal bifurcations. A hyperbolic polycycle with nine edges is constructed whose semilocal bifurcation in an open set of nine-parameter families has a functional invariant.

This paper addresses the problem of a homogeneous ball rolling on the inner surface of a circular cylinder in a field of gravity parallel to its axis. It is assumed that the ball rolls without slipping on the surface of the cylinder, and that the cylinder executes plane-parallel motions in a circle perpendicular to its symmetry axis. The integrability of the problem by quadratures is proved. It is shown that in this problem the trajectories of the ball are quasiperiodic in the general case, and that an unbounded elevation of the ball is impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonances at which the ball moves on average downward with constant acceleration.

In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom in small neighborhoods of singular points having the type ``universal unfolding of $A_n$ singularity'', $n\geqslant 1$ (local singularities), as well as in small neighborhoods of compact orbits containing such singular points (semilocal singularities). We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semiglobal classification). These corank-one singularities (local, semilocal and semiglobal ones) are structurally stable. It turns out that all integrable systems are symplectically equivalent near their singular points of this type, thus there are no local symplectic invariants. A complete semilocal (respectively, semiglobal) symplectic invariant of the singularity is given by a tuple of $n-1$ (respectively $n-1+\ell$) real-analytic function germs in $n$ variables, where $\ell$ is the number of connected components of the complement of the singular orbit in the fiber. The case $n=1$ corresponds to nondegenerate singularities (of elliptic and hyperbolic types) of one-degree-of-freedom Hamiltonians; their symplectic classifications were known. The case $n=2$ corresponds to parabolic points, parabolic orbits and cuspidal tori.

This paper addresses the spatial restricted elliptic problem of three bodies (material points) gravitating toward each other under Newton’s law of gravitation. The eccentricity of the orbit of the main attracting bodies is assumed to be small, and nonlinear oscillations of a passively gravitating body near a Lagrangian triangular libration point are studied. It is assumed that in the limiting case of the circular problem the ratio of the frequency of rotation of the main bodies about their common center of mass to the value of one of the frequencies of small linear oscillations of the passive body is exactly equal to three. A detailed analysis is made of two different particular cases of influence of the three-dimensionality of the problem on the characteristics of nonlinear oscillations of the passive body.

We study two-dimensional Riemannian metrics which are superintegrable in the class of integrals polynomial in momenta. The study is based on our main technical result, Theorem 2, which states that the Poisson bracket of two integrals polynomial in momenta is an algebraic function of the integrals and of the Hamiltonian. We conjecture that twodimensional superintegrable Riemannian metrics are necessarily real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. A small modification of the arguments, discussed in the paper, provides a method to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show that the metrics constructed by K. Kiyohara [9], which admit irreducible integrals polynomial in momenta, of arbitrary high degree $k$, are not superintegrable and in particular do not admit nontrivial integrals polynomial in momenta, of degree less than $k$. This result solves Conjectures (b) and (c) explicitly formulated in [4].

Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex, closed $C^3$ hypersurface of that hyperplane, with an everywhere nondegenerate second fundamental form. In this paper, we prove that there exist $C^2$ convex cones with billiard trajectories that undergo infinitely many reflections in finite time. We also provide an estimation of the number of reflections for billiard trajectories inside elliptic cones in $\mathbb{R}^3$ using two first integrals.

In this paper we consider nonsingular Morse – Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold admitting such flows is either a lens space, or a connected sum of a lens space with a projective space, or Seifert manifolds with base sphere and three special layers. A complete topological classification of the described flows is obtained and the number of their equivalence classes on each admissible manifold is calculated.

We consider the problem of existence of forced oscillations on a Riemannian manifold, the metric on which is defined by the kinetic energy of a mechanical system. Under the assumption that the generalized forces are periodic functions of time, we find periodic solutions of the same period. We present sufficient conditions for the existence of such solutions, which essentially depend on the behavior of geodesics on the corresponding Riemannian manifold.

We present some new Poisson bivectors that are invariants by the Clebsch system flow. Symplectic integrators on their symplectic leaves exactly preserve the corresponding Casimir functions, which have different physical meanings. The Kahan discretization of the Clebsch system is discussed briefly.