# Applied and Computational Mathematics Seminar

## Seminar Details

- Speaker:
- Peter Lynch
- Affiliation:
- University College Dublin
- Title:
- Integrable elliptic billiards and ballyards
- Time:
- 1PM Monday, 9 September 2019
- Location:
- SCN 1.25, O'Brien Centre for Science (North)

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.

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