Stability of Thomson’s Configurations of Vortices Ona Sphere
نویسندگان
چکیده
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated. 1 A linear stability of Thomson’s configurations of vortices on a sphere Motion equations of a system of N point vortices on a sphere can be presented in the Hamiltonian form [1] with the Poisson bracket {φi, cos θk} = δik RΓi (1) and the Hamiltonian H = − 1 8π N ∑′ i, k=1 ΓiΓk ln ( 4R sin γik 2 ) , cos γik = cos θi cos θk + sin θi sin θk cos(φk − φi) , (2) REGULAR AND CHAOTIC DYNAMICS V.5, No. 2, 2000 Received January 17, 2000 AMS MSC 76C05, 58F10
منابع مشابه
Point vortices on a sphere: Stability of relative equilibria
In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the ~integrable! case of three vortices. The system under consideration is SO~3! invariant; the associated momentum map generated by this SO~3! symmetry is equivariant and cor...
متن کاملNonlinear Stability of a Latitudinal Ring of Point-Vortices on a Nonrotating Sphere
We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude θ and of the number of vortices N . By using the integrals of motion, we view the system in a suitable corotating frame where the polygonal vortex configuration is at rest. Then after a...
متن کاملM ar 2 00 5 DYNAMICS OF THREE VORTICES ON A PLANE AND A SPHERE — III Noncompact case . Problems of collaps and scat - tering
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 realization. We completed the bifurcation analysis and investigated for collinear and Thomson's configurations the dependence of stability in linear approximation a...
متن کاملM ar 2 00 5 DYNAMICS AND STATICS OF VORTICES ON A PLANE AND A SPHERE —
In the present paper a description of a problem of point vortices on a plane and a sphere in the " internal " variables is discussed. The Hamiltonian equations of motion of vortices on a plane are built on the Lie–Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partia...
متن کامل