Generalization of the Goryachev-Chaplygin Case

In this paper we present a generalization of the Goryachev-Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics' equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.Comment: 10 page

Euler-Poisson Equations and Integrable Cases

In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev-Chaplygin cases of Euler-Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.Comment: 24 pages, 27 figure

Generalization of the Goraychev--Chaplygin Case

In this paper we present a generalization of the Goraychev--Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics' equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.Comment: 15 page

Superintegrable systems on sphere

We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole (the Poincar\'e--Appell system) and addition of a more complicated field, which itself is a generalization of the MICZ-system. The mentioned systems are integrable -- in fact, superintegrable. The latter is due to the vector integral, which is analogous to the Laplace--Runge--Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.Comment: 14 pages, 2 figure

On the spectral stability of kinks in 2D Klein-Gordon model with parity-time-symmetric perturbation

In a series of recent works by Demirkaya et al. stability analysis for the static kink solutions to the 1D continuous and discrete Klein-Gordon equations with a $\mathcal{PT}$-symmetric perturbation has been analysed. We consider the linear stability problem for the static kink in 2D Klein-Gordon field taking into account spatially localized $\mathcal{PT}$-symmetric perturbation. The perturbation is in the form of viscous friction, which does not affect the static solutions to the unperturbed Klein-Gordon equation. Small dynamic perturbation around the static kink solution is considered to formulate the linear stability problem. The effect of the small perturbation on the solutions to the corresponding eigenvalue problem is analysed. The main result is presented in the form of a theorem describing the behavior of the eigenvalues corresponding to the extended and localised eigenmodes as the functions of the perturbation parameter

On the History of the Development of the Nonholonomic Dynamics

The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.Comment: 5 page

The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics

In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of body of revolution and the motion of round disk with sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi integrable and have additional integrals and invariant measure. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. Here we consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions. We also consider the generalized problem of rolling of dynamically nonsymmetric Chaplygin ball. The results of studies are presented as tables that describe the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure in the considered problems.Comment: 24 pages, 11 figures, 2 table

Absolute and relative choreographies in the problem of point vortices moving on a plane

We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.Comment: 11 pages, 6 figure

Two-body problem on a sphere. Reduction, stochasticity, periodic orbits

We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.Comment: 15 pages, 6 figure

Reduction and chaotic behavior of point vortices on a plane and a sphere

We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.Comment: 10 pages, 3 figure