# Physical Versus Mathematical Billiards: From Regular Dynamics to Chaos and Back

## TL;DR

This paper explores how transitioning from idealized point-particle billiards to physical finite-sized particles can dramatically alter the system's dynamics, causing chaos to appear or disappear unpredictably.

## Contribution

It demonstrates that changing from mathematical to physical billiards can induce complex, sometimes abrupt, shifts in dynamical behavior, including chaos emergence or suppression, depending on particle size.

## Key findings

- Non-chaotic billiards can become chaotic with finite particle size
- Chaotic systems can become regular when particle size changes
- Dynamics may change multiple times as particle size varies

## 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 non-chaotic 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 completely (globally). These results are somewhat unexpected because for all standard examples of billiards their dynamics remains absolutely the same after transition from a point particle to a finite size ("physical") particle. Moreover we show that a character of dynamics may change several times when the size of particle is increasing.