The characteristic exponents of the falling ball model

Nandor Simanyi

doi:10.1007/BF02517897

paper

The characteristic exponents of the falling ball model

arXiv:math/9604233 · doi:10.1007/BF02517897

Abstract

We study the characteristic exponents of the Hamiltonian system of $n$ ( $\geq 2$ ) point masses $m_{1}, \dots, m_{n}$ freely falling in the vertical half line ${q ∣ q \geq 0}$ under constant gravitation and colliding with each other and the solid floor $q = 0$ elastically. This model was introduced and first studied by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic (Lyapunov) exponents of the above dynamical system are nonzero, provided that $m_{1} \geq \dots \geq m_{n}$ (i. e. the masses do not increase as we go up) and $m_{1} \neq = m_{2}$ .

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The flow of two falling balls mixes rapidly