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On Cayley conditions for billiards inside ellipsoids (1211.6557v1)

Published 28 Nov 2012 in math.DS and nlin.SI

Abstract: All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of $\Rsetn$ are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard trajectories verify certain algebraic conditions. Cayley found them in the planar case. Dragovi\'{c} and Radnovi\'{c} generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several remarkable algebraic relations between caustic parameters and ellipsoidal parameters that give rise to nonsingular periodic trajectories.

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