Billiards of variable configuration and billiards with slippage in Hamiltonian geometry and topology

Abstract

A class of billiards is found, the geometry of which can change with a change in the energy of a ball moving on a «billiard table». Such billiards are called force or evolutionary. They make it possible to implement important integrable Hamiltonian systems (with two degrees of freedom) on the entire phase 4-dimensional space of the system at once. That is, simultaneously on all regular isoenergetic 3-dimensional surfaces. The author and V.V. Vedyushkina proved that force billiards implement the Euler and Lagrange integrable cases in the dynamics of a heavy body in three-dimensional space. It is found that these two well-known systems «billiard equivalent», although they have integrals of different degrees – quadratic (Euler) and linear (Lagrange).