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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Relative equilibria of point vortices. (Aref Memor
ial Lecture) - Brns\, M (Technical University of D
enmark)
DTSTART;TZID=Europe/London:20120724T164500
DTEND;TZID=Europe/London:20120724T173000
UID:TALK39024AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/39024
DESCRIPTION:A relative equilibrium of a system of point vortic
es is a configuration which rotates with constant
angular velocity around its centre of vorticity. I
t is easy to write down the equations for the vort
ex positions and many simple configurations with s
ymmetry are known. Several asymmetric states have
been found numerically\, including some surprising
ones with some of the vortices being very close.
Very little is known analytically about the genera
l problem.\n \nHere we consider the case where the
vortices are identical and placed on two perpendi
cular lines which we choose to be the axes of a co
ordinate system. We define two polynomials p(z) an
d q(z) whose roots are the vortex positions on eac
h line in the complex plane\, and derive a differe
ntial equation for p for given q. We discuss how t
he general solution to the differential equation r
elates 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 degr
ee n\, it must have at least n-m+2 real roots. For
m=2 this is a complete characterisation\, and we
obtain an asymptotic result for the location of th
e two vortices on the imaginary axis as the number
of vortices on the real axis tends to infinity.\n
\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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