Estimating Lyapunov exponents in billiards.

Area of Science:

• Physics
• Statistical Mechanics
• Dynamical Systems

Background:

• Dynamical billiards are fundamental models for chaotic Hamiltonian systems.
• Understanding chaotic dynamics is crucial for various physics applications.
• Lyapunov exponents quantify the rate of chaos in dynamical systems.

Purpose of the Study:

• To investigate the relationship between Lyapunov exponents and phase space volume in dynamical billiards.
• To explore the influence of external parameters, specifically magnetic fields, on chaotic dynamics.
• To extend existing formalisms for calculating Lyapunov exponents.

Main Methods:

• Analysis of phase space volume arguments for Lyapunov exponent estimation.
• Extension of the Dellago, Posch, and Hoover formalism.
• Implementation of a software tool for Lyapunov exponent calculation in billiards with magnetic fields.

Main Results:

• A clear inverse proportionality was found between the leading Lyapunov exponent and the chaotic phase space volume across diverse billiards.
• The study successfully extended the theoretical framework to incorporate external magnetic fields.
• A functional software implementation for calculating Lyapunov exponents under magnetic fields was developed.

Conclusions:

• The inverse relationship between Lyapunov exponent and chaotic phase space volume appears to be a general principle in dynamical billiards.
• The extended formalism and software provide valuable tools for studying chaotic Hamiltonian systems, particularly under magnetic field influence.