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

Area of Science:

• * Mathematical Physics
• * Dynamical Systems Theory

Background:

• * Standard mathematical billiards involve point particles with elastic boundary reflections.
• * Physical billiards use finite-size hard spheres, differing from the idealized point particle model.

Purpose of the Study:

• * To investigate the dynamical transitions when moving from mathematical to physical billiards.
• * To analyze how finite particle size affects the chaotic or nonchaotic nature of billiard systems.

Main Methods:

• * Theoretical analysis of billiard dynamics with finite-size particles.
• * Examination of phase portrait changes based on particle radius variations.

Main Results:

• * Finite particle size can induce chaos in nonchaotic systems and vice versa.
• * Transitions between chaotic and nonchaotic dynamics can be soft (gradual) or hard (abrupt).
• * The character of billiard dynamics can change multiple times as particle size increases.

Conclusions:

• * The transition from mathematical to physical billiards introduces complex, often unexpected, dynamical behaviors.
• * Finite particle size fundamentally alters billiard dynamics, challenging assumptions of stability in standard models.