Reducing the Degree of Integrals of Hamiltonian Systems by Using Billiards

Abstract

In the theory of integrable Hamiltonian systems with two degrees of freedom, widely known are integrable systems having integrals of high degrees, namely, 3 and 4. Examples are the Kovalevskaya system and its generalizations—the Kovalevskaya–Yehia system and the Kovalevskaya system on the Lie algebra so(4) the Goryachev–Chaplygin–Sretensky, Sokolov, and Dullin–Matveev systems. It is shown that, at a number of isoenergy 3-surfaces, the third and fourth degrees of integrals of these systems can be reduced by using integrable billiards bounded by arcs of confocal quadrics. Moreover, the integrals of degree 3 and 4 are reduced to the same canonical quadratic integral on a billiard.