  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.
