No-slip billiards with particles of variable mass distribution

Area of Science:

• Mathematical Physics
• Dynamical Systems Theory
• Chaos Theory

Background:

• Mathematical billiards, particularly those with specular (frictionless) collisions, are well-studied for their diverse dynamical behaviors, ranging from regular to chaotic.
• Understanding these dynamics is crucial for various fields, including statistical mechanics and quantum chaos.

Purpose of the Study:

• To investigate the influence of particle mass distribution on the dynamics of mathematical billiards with no-slip (rough) collisions.
• To explore whether varying mass distribution can induce a rich spectrum of dynamics comparable to or exceeding that of geometric variations.
• To identify novel dynamical behaviors, such as full measure periodic billiards, in systems with non-uniform particle mass.

Main Methods:

• Studied three two-parameter families of billiards, varying both table geometry and particle mass distribution.
• Analyzed systems under no-slip collision conditions, extending the concept of standard specular billiards.
• Employed mathematical analysis and computational simulations to observe and classify dynamical behaviors.

Main Results:

• Demonstrated that altering the mass distribution of the colliding particle can lead to markedly divergent dynamical behaviors, independent of table geometry.
• Observed a rich spectrum of dynamics, including integrable, chaotic, and unusual behaviors, solely by manipulating particle mass.
• Identified specific parameter regimes yielding full measure periodic billiards, a phenomenon previously conjectured to be absent in standard Euclidean billiards.

Conclusions:

• Particle mass distribution is a significant factor in determining the dynamical complexity of mathematical billiards, offering a new dimension for controlling system behavior.
• The findings expand the understanding of billiard dynamics, suggesting that rough collisions combined with variable mass distributions can generate novel and complex behaviors.
• The discovery of potential full measure periodic billiards opens new avenues for theoretical and computational research in dynamical systems.