Topological Analysis of an Elliptic Billiard in a Fourth-Order Potential Field

A planar billiard is considered in an elliptic domain in the case where a polynomial potential of the fourth degree acts on a material point. This dynamical system always admits the first integral called the total energy, the Hamiltonian of the system. Under additional conditions imposed on the potential to guarantee for the system the existence of another first integral, which is independent of the Hamiltonian, the system becomes completely Liouville integrable. The paper presents a topological analysis of the corresponding Liouville foliation of this system; namely, bifurcation diagrams are constructed and Fomenko-Zieschang invariants are calculated.