Peter Lynch (University College Dublin)

will speak on

Integrable Elliptic Billiards & Ballyards

Time: 1:00PM
Date: Mon 9th September 2019
Location: Seminar Room SCN 1.25 [map]

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Abstract: The billiard problem concerns a point particle moving freely in a
region of the horizontal plane bounded by a closed curve $\Gamma$,
and reflected at each impact with $\Gamma$. The region is called
a `billiard', and the reflections are specular: the angle of
reflection equals the angle of incidence. We review the dynamics
in the case of an elliptical billiard. In addition to conservation
of energy, the quantity $L_1 L_2$ is an integral of the motion,
where $L_1$ and $L_2$ are the angular momenta about the two foci.

We can regularize the billiard problem by approximating the
flat-bedded, hard-edged surface by a smooth function. We then
obtain solutions that are everywhere continuous and differentiable.
We call such a regularized potential a `ballyard'. A class of
ballyard potentials will be defined that yield systems that are
completely integrable. We find a new integral of the motion that
corresponds, in the billiards limit $N\to\infty$, to $L_1 L_2$.

Just as for the billiard problem, there is a separation of the
orbits into boxes and loops. The discriminant that determines the
character of the solution is the sign of $L_1 L_2$ on the major
axis.

(This talk is part of the Applied and Computational Mathematics series.)

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