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SUMMARY:Relative equilibria of point vortices. (Aref Memorial Lecture) - B
 rns\, M (Technical University of Denmark)
DTSTART:20120724T154500Z
DTEND:20120724T163000Z
UID:TALK39024@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:A relative equilibrium of a system of point vortices is a conf
 iguration which rotates with constant angular velocity around its centre o
 f vorticity. It is easy to write down the equations for the vortex positio
 ns and many simple configurations with symmetry are known. Several asymmet
 ric states have been found numerically\, including some surprising ones wi
 th some of the vortices being very close. Very little is known analyticall
 y about the general problem.\n \nHere we consider the case where the vorti
 ces are identical and placed on two perpendicular lines which we choose to
  be the axes of a coordinate system. We define two polynomials p(z) and q(
 z) whose roots are the vortex positions on each line in the complex plane\
 , and derive a differential equation for p for given q. We discuss how the
  general solution to the differential equation relates to physical vortex 
 configurations. The main result is that if q has m solutions symmetrically
  placed relative to the real axis and p is of degree n\, it must have at l
 east n-m+2 real roots. For m=2 this is a complete characterisation\, and w
 e obtain an asymptotic result for the location of the two vortices on the 
 imaginary axis as the number of vortices on the real axis tends to infinit
 y.\n\n
LOCATION:Seminar Room 1\, Newton Institute
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