|
Эта публикация цитируется в 17 научных статьях (всего в 17 статьях)
Euler Configurations and Quasi-Polynomial Systems
A. Albouya, Y. Fub a Astronomie et Systémes Dynamiques, IMCCE, 77, av. Denfert-Rochereau, Paris 75014, France
b Purple Mountain Observatory, 2 West Beijing Road, Nanjing 210008, P. R. China
Аннотация:
Consider the problem of three point vortices (also called Helmholtz' vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to $1/r$, where $r$ is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to $r^b$, for any $b<0$. For $0<b<1$, the optimal upper bound becomes 5. For positive vorticities and any b<1, there are exactly 3 collinear normalized relative equilibria. The case $b=-2$ of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.
Ключевые слова:
relative equilibria, point vortex, real solutions.
Поступила в редакцию: 12.07.2006 Принята в печать: 14.09.2006
Образец цитирования:
A. Albouy, Y. Fu, “Euler Configurations and Quasi-Polynomial Systems”, Regul. Chaotic Dyn., 12:1 (2007), 39–55
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd610 https://www.mathnet.ru/rus/rcd/v12/i1/p39
|
|