The reduction of the degree of integrals of hamiltonian systems with the help of billiards

Abstract

In the theory of integrable Hamiltonian systems with two degrees of freedom there are widely known integrable systems whose integrals have a high degree, namely 3 and 4: the Kovalevskaya system and its generalizations - the Kovalevskaya - Yahya system and the Kovalevskaya system on the Lie algebra so(4), Goryachev-Chaplygin-Sretensky, Sokolov and Dullin-Matveyev. The article shows that using integrable billiards bounded by arcs of confocal quadrics decreases the degree of integrals 3 and 4 of these systems fo some isoenergy 3-surfaces. Moreover, the integrals of degree 3 and 4 reduce to the same canonical quadratic integral on billiards.