Vortex motion |
The N vortex system has 4 classical integrals of motion:
two integrals for the center of mass
,
one integral for the angular momentum
and one integral for the energy H.
The center of mass is invariant because
every term in
appears twice with opposite sign.
To see the angular momentum invariance, define
Because I and
are real,
holds and
Because H,I,J have pairwise non-vanishing Poisson brackets,
a 3 vortex system is explicitly integrable.
The threshold for chaotic behavior starts with N=4.
Already the restricted 4-vortex system is non-integrable. It is
the limiting case, where one of the vortices does not contribute to the
vorticity.
If the vorticities are positive and ,
the solution
of the vortex flow exists for all times
because
defines a compact
set and because H=const prevents collisions.
If takes both signs the existence and uniqueness problem is nontrivial and catastrophes are possible: the 3-vortex situation and for example leads to a collapse at time : if a_{i} are the lengths of the triangle formed by z_{1},z_{2},z_{3}and A is the triangle area, one has . The ratios of the triangle lengths are conserved implying so that .
Kolmogorov-Arnold- Moser
theory implies that there exists always
quasi-periodic motion in the phase space of a N-vortex system.
A vortex system of four or more vortices shows therefore a mixture
of stable and unstable behavior.
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© 2000, Oliver Knill , dynamical-systems.org |