Volume 17, Number 5
Volume 17, Number 5, 2012
Demina M. V., Kudryashov N. A.
Point Vortices and Classical Orthogonal Polynomials
Abstract
Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.

Kurakin L. G., Ostrovskaya I. V.
Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle
Abstract
A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius $R_0$ outside a circular domain of radius $R$ is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter $q = R^2/R_0^2$.

Fusco G., Negrini P., Oliva W. M.
Stationary Motion of a Selfgravitating Toroidal Incompressible Liquid Layer
Abstract
We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian selfattraction. Under the assumption of infinitesimal thickness of the stratum we show the existence of stationary motions during which the stratum is approximately a round torus (with radii $r$, $R$ and $R \gg r$) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helixlike trajectory around the circumference of radius $R$ that connects the centers of the cross sections of the torus.

Howard J. E., Morozov A. D.
A Simple Reconnecting Map
Abstract
Generalized standard maps of the cylinder for which the rotation number is a rational function (a combination of the Fermi and Chirikov rotation functions) are considered. These symplectic maps often have degenerate resonant zones, and we establish two types resonance bifurcations: "loops" and "vortex pairs". Both the border of chaos and the existence of the chaotic web are discussed. Finally the transition to global chaos for a generalized map is considered.

Dragović V., Gajic B.
On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations
Abstract
It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for $B \ne 0$ is not an algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Fourdimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie–Poisson bracket.

Tsiganov A. V.
On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems
Abstract
We discuss a Poisson structure, linear in momenta, for the generalized nonholonomic Chaplygin sphere problem and the $LR$ Veselova system. Reduction of these structures to the canonical form allows one to prove that the Veselova system is equivalent to the Chaplygin ball after transformations of coordinates and parameters.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
The Bifurcation Analysis and the Conley Index in Mechanics
Abstract
The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.
