Physical versus mathematical billiards: From regular dynamics to chaos and back.

Abstract

In standard (mathematical) billiards, a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite-size hard sphere moves at the same billiard table, virtually anything may happen. Namely, a nonchaotic billiard may become chaotic and vice versa. Moreover, both these transitions may occur softly, i.e., for any (arbitrarily small) positive value of the radius of a physical particle, as well as by a "hard" transition when radius of the physical particle must exceed some critical strictly positive value. Such transitions may change a phase portrait of a mathematical billiard locally as well as globally. These results are somewhat unexpected because for standard examples of billiards, their dynamics remains absolutely the same after replacing a point particle by a finite-size ("physical") particle. Moreover, we show that a character of dynamics may change several times when the size of particle is increasing.