NDLI: Euler configurations and quasi-polynomial systems

Content Provider	Springer Nature Link
Author	Albouy, A. Fu, Y.
Copyright Year	2007
Abstract	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.
Starting Page	39
Ending Page	55
Page Count	17
File Format	PDF
ISSN	15603547
Journal	Regular and Chaotic Dynamics
Volume Number	12
Issue Number	1
e-ISSN	14684845
Language	English
Publisher	Nauka/Interperiodica
Publisher Date	2007-01-01
Publisher Place	Moscow
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	relative equilibria point vortex real solutions Dynamical Systems and Ergodic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistical and Nonlinear Physics Mathematical Physics Mechanical Engineering Modeling and Simulation

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