# Estimating Lyapunov exponents in billiards

## TL;DR

This paper investigates how to estimate Lyapunov exponents in billiards using phase space volume arguments, revealing an inverse proportionality in mixed phase spaces and extending existing methods to include magnetic fields.

## Contribution

It introduces a phase space volume-based estimation method for Lyapunov exponents in billiards and extends existing formalism to account for magnetic fields.

## Key findings

- Lyapunov exponent inversely proportional to chaotic phase space volume
- Extended formalism to include magnetic fields
- Provided software implementation for calculations

## Abstract

Dynamical billiards are paradigmatic examples of chaotic Hamiltonian dynamical systems with widespread applications in physics. We study how well their Lyapunov exponent, characterizing the chaotic dynamics, and its dependence on external parameters can be estimated from phase space volume arguments, with emphasis on billiards with mixed regular and chaotic phase spaces. We show that in the very diverse billiards considered here the leading contribution to the Lyapunov exponent is inversely proportional to the chaotic phase space volume, and subsequently discuss the generality of this relationship. We also extend the well established formalism by Dellago, Posch, and Hoover to calculate the Lyapunov exponents of billiards to include external magnetic fields and provide a software implementation of it.