Evolutionary force billiards

Abstract

A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are "billiard-equivalent", despite the fact that the former one is square integrable, and the latter one allows a linear integral.

Billiards and integrable systems

“…Systems of the Lagrange top can have -depending on the value of the area integral, the relation between the moments of inertia, and the choice of the potential -four types of bifurcation diagram (see [6]) and precisely five types of symplectic 4-sheets. For all of them we found force billiards [58]. We show one of these in Fig.…”

“…Theorem 16 ([57], [58]). An integrable force billiard whose support is homeomorphic to an ellipsoid realizes (in the sense of Liouville equivalence) the integrable Euler case on the whole of the phase manifold M 4 g at the same time, apart from the singular levels of energy, that is, on all regular isoenergy 3-surfaces for all regular values of g and h.…”

“…15. Theorem 17 ([57], [58]). On each regular symplectic 4-sheet M 4 g the Lagrange integrable case can be realized (in the sense of Liouville equivalence) by one of the five force billiards constructed in [57].…”

“…Theorem 18 ([57], [58]). The above deformation of confocal billiards into circular ones transforms the force billiard realizing the Euler case into another force billiard, the full system of Liouville foliations for which coincides with the full system of foliations for the Lagrange case (obtained by combining all three types of isoenergy 3-surfaces).…”

“…In § 3 we describe in detail the class of evolutionary billiards introduced recently [57], [58] and present result on modelling, by means of such systems, the topology of the (whole) Euler and Largange tops (so that all non-singular foliations on the symplectic phase space M 4 of the system are realized by a single evolutionary billiard), the Joukowsky system, and the Kovalevskaya top [59].…”