Vol 295, No 1 (2016)

We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group G₂ with nonnull orbits and electromagnetic fields possess the same symmetry. For the fields with such symmetries, we describe the structures of the so-called non-dynamical degrees of freedom, whose presence, just as the presence of a cosmological constant, changes (in a strikingly similar way) the vacuum and electrovacuum dynamical equations and destroys their well-known integrable structures. We find modifications of the known reduced forms of Einstein–Maxwell equations, namely, the Ernst equations and the self-dual Kinnersley equations, in which the presence of non-dynamical degrees of freedom is taken into account, and consider the following subclasses of fields with different non-dynamical degrees of freedom: (i) vacuum metrics with cosmological constant; (ii) space–time geometries in vacuum with isometry groups G₂ that are not orthogonally transitive; and (iii) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. For each of these classes of fields, in the case when the two-dimensional metrics on the orbits of the isometry group G₂ are diagonal, all field equations can be reduced to one nonlinear equation for one real function α(x1, x2) that characterizes the area element on these orbits. Simple examples of solutions for each of these classes are presented.

This paper provides a detailed description of various reduction schemes in rigid body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the cases already found in order and to obtain new generalizations of the Kovalevskaya case to e(3). Note that the indicated reduction allows one to obtain in a natural way some singular additive terms that were proposed earlier by D.N. Goryachev.

The paper deals with nearly integrable multidimensional a priori unstable Hamiltonian systems. Assuming the Hamilton function is smooth and time-periodic, we study perturbations that are trigonometric polynomials in the “angle” variables in the first approximation. For a generic system in this class, we construct a trajectory whose projection on the space of slow variables crosses a small neighborhood of a strong resonance. We also estimate the speed of this crossing.

We present an abstract KAM theorem adapted to space-multidimensional Hamiltonian PDEs with smoothing nonlinearities. The main novelties of this theorem are the following: (i) the integrable part of the Hamiltonian may contain a hyperbolic part and, as a consequence, the constructed invariant tori may be unstable; (ii) it applies to singular perturbation problems. In this paper we state the KAM theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem.

We present the results of theoretical and experimental investigations of the motion of a spherical robot on a plane. The motion is actuated by a platform with omniwheels placed inside the robot. The control of the spherical robot is based on a dynamic model in the nonholonomic statement expressed as equations of motion in quasivelocities with indeterminate coefficients. A number of experiments have been carried out that confirm the adequacy of the dynamic model proposed.

We consider a self-similar piston problem in which stresses on the boundary of a half-space are changed instantaneously. The half-space is filled with a Prandtl–Reuss medium in a uniform stressed state. It is assumed that the formation of shock waves is possible in the medium. We prove the existence of a solution to the problem in the cases when two or all three stress components are changed at the initial moment.

The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis O_x1 of periodically alternating M elastic and M viscoelastic layers parallel to the plane O_x2x3. It is shown that the spectrum of the boundary value problem is the union of roots of M equations. The asymptotic behavior of the spectrum of the problem as M → ∞ is analyzed; in particular, it is proved that not all sequences of eigenvalues of the original (prelimit) problem converge to eigenvalues of the corresponding homogenized (limit) problem.

For a geodesic (or magnetic geodesic) flow, the problem of the existence of an additional (independent of the energy) first integral that is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.

The asymptotic method of global instability developed by A.G. Kulikovskii is an effective tool for determining the eigenfrequencies and stability boundary of one-dimensional or multidimensional systems of sufficiently large finite length. The effectiveness of the method was demonstrated on a number of one-dimensional problems; and since the mid-2000s, this method has been used in aeroelasticity problems, which are not strictly one-dimensional: such is only the elastic part of the problem, while the gas flow occupies an unbounded domain. In the present study, the eigenfrequencies and stability boundaries predicted by the method of global instability are compared with the results of direct calculation of the spectra of the corresponding problems. The size of systems is determined starting from which the method makes a quantitatively correct prediction for the stability boundary.