Related papers: Symmetric Relative Equilibria in the Four-Vortex P…
We consider the Newtonian 5-body problem in the plane, where 4 bodies have the same mass m, which is small compared to the mass M of the remaining body. We consider the (normalized) relative equilibria in this system, and follow them to the…
There is an explicit resolution of the Poisson reduction of four planar point vortices, in the case that three of the vortex strengths are equal and the total vorticity is zero. The resolution, a Hamiltonian system on a unified symplectic…
This paper addresses the question of existence of (not necessarily self-similar) solutions to the 4-vortex problem that lead to total or partial collision. We begin by showing that energy considerations alone imply that, for the general…
Giving a new form of the vortex mode equation by a proper change of parameter, our aim is to analyze the point and contact symmetries of the new equation. Fundamental invariants and a form of general solutions of point transformations along…
In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the…
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…
We consider the $n$ body problem defined on surfaces of constant positive curvature. For the 5 and 7 body problem in a collinear symmetric configuration we obtain initial positions which lead to relative equilibria. We give explicitly the…
The motion of two pairs of counter-rotating point vortices placed in a uniform flow past a circular cylinder is studied analytically and numerically. When the dynamics is restricted to the symmetric subspace---a case that can be realized…
We give an exact quantitative solution for the motion of three vortices of any strength, which Poincar\'e showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is…
Based on the analysis of the velocity gradient tensor, we investigate in this paper the physical interpretation and limitations of four vortex criteria: $\omega$, $Q$, $\varDelta$ and $\lambda_{ci}$, and reveal the actual physical meaning…
A system of three point vortices in an unbounded plane has a special family of self-similarly contracting or expanding solutions: during the motion, vortex triangle remains similar to the original one, while its area decreases (grows) at a…
We prove that for some potentials (including the Newtonian one, and the potential of Helmholtz vortices in the plane) relative equilibria consisting of two homothetic regular polygons of arbitrary size can only occur if the masses at each…
Relative equilibria on a rotating meridian on $\mathbb{S}^2$ in equal-mass three-body problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria. Almost all isosceles triangles,…
The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient…
In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and a scattering of vortices and obtained the conditions of its…
For the planar four-vortex problem, we show that there are finitely many stationary configurations consisting of equilibria, rigidly translating configurations, relative equilibria (uniformly rotating configurations) and collapse…
The finiteness problem of stationary configurations for the planar five-vortex problem is considered in this paper. The numbers of equilibria and rigidly translating configurations are shown to be at most 6 and 24 respectively. The numbers…
Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central…
Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of…
For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is…