"Billiards" by Marie-Claude Arnaud

You play billiard in a smooth and strictly convex table ... which kinds of trajectories will you observe? We will explain: - Why mathematicians began to look at this problem? - How to model this? In particular, we will see that the billiard map can be seen as a map of a (bounded) cylinder. - How to find trajectories? We will see that a lot of orbits are maximizers of a certain function and that this function is the length of the trajectory. We will prove the existence of periodic orbits and speak of caustics, that are curves that sometimes appear if your ball has an infinite trajectory... Prerequisites : norm, scalar product, angle, bisector, geometry of the triangle, area (for example area of a parallelogram), derivative, tangent and normal vector to a planar curve, linear momentum and kinetic energy.

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