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DCDS-B

The Hamiltonian system governing
$N$-interacting particles
constrained to lie on a
closed planar curve are derived.
The problem
is formulated in detail for the case of logarithmic
(point-vortex) interactions.
We show that when the curve is circular with radius
$ R $, the system is completely integrable for
all particle strengths $ \Gamma _ \beta $, with particle $ \Gamma _ \beta $
moving with frequency $ \omega _ \beta = (\Gamma - \Gamma _ \beta )/4 \pi R^2 $,
where $ \Gamma = \sum^{N}_{\alpha=1} \Gamma _ \alpha $
is the sum of the strengths of all the particles.
When all the particles have equal strength, they move
periodically around the circle keeping
their relative distances fixed.
When not all the strengths are equal, two or more
of the particles collide in finite time.
The diffusion of a neutral particle (i.e. the problem of 1D mixing) is examined.
On a circular curve,
a neutral particle moves
uniformly with frequency
$ \Gamma / 4 \pi R^2 $.
When the curve is not perfectly circular,
for example when given a sinusoidal perturbation,
or when the particles
move on concentric circles with different
radii,
the particle dynamics is considerably more complex, as shown numerically
from an examination of power spectra
and collision diagrams. Thus, the circular constraint appears to
be special in that it induces completely integrable dynamics.

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