Vortex motion

The motion of N vortices with voriticity located at points zi in the complex plane is determined by the differential equations

This is a Hamiltonian system

for the variables xi+i yi=zi. The Hamiltonian function or energy is

Using the notation one can write

the variables are canonically conjugated. The phase space of the system is , where is the collision set.

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

is purely imaginary so that . The energy is invariant because . These integrals come from symmetry: the translational, rotational and time invariance of the differential equations. With the Poisson bracket

for two smooth functions , the notation and the scalar product on one has

and the equations of motion are or

The invariance of the integrals means . Other integrals are obtained by combining known ones. For example is an integral.

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 ai are the lengths of the triangle formed by z1,z2,z3and 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