New examples of systems of the Kowalevski type

Abstract

A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two theta-functions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure.

Systems of Kowalevski type and discriminantly separable polynomials

“…In the so-called fundamental Kowalevski equation (see formula (10) below, and also [20], [19], [13]) Q(w, x 1 , x 2 ) = 0, the polynomial Q(w, x 1 , x 2 ) appeares to be an example of a member of the family, as it was shown in [8] (Theorem 3). Moreover, all main steps of the Kowalevski integration now follow as easy and transparent logical consequences of the theory of discriminantly separable polynomials.…”

“…Thus we call the members of that class -systems of the Kowalevski type. A relationship with the discriminantly separable polynomials gives us possibility to perform an effective integration procedure, and to provide an explicit integration formulae in the theta-functions, in general, associated with genus two curves, as in the original case of Kowalevski. The first examples of such systems have been constructed in [10]. Let us point out here one very important moment regarding the Kowalevski top and all the systems of Kowalevski type: the main issue in integration procedures is related to the elliptic curves Γ 1 , Γ 2 and the two-valued groups related to these elliptic curves, although, as we know, the final part of integration of the Kowalevski is related to a genus two curve.…”

Role of discriminantly separable polynomials in integrable dynamical systems

“…Here and denote roots of equation (1) as a quadratic equation in s. In [5] we developed theory of the systems of the Kovalevsky type and explained in detail how Kowalevski's integration procedure can be applied on a whole class of systems. In [3] we presented few new examples of such systems. It is a characteristic property of a whole class of Kowalevski type systems that they can be explicitly integrated in theta functions of genus two.…”

“…After equating the square of from the relation (10) with product from equations (9) one can check that the variables of the Sokolov's system satisfy the following identity: (12) To see that Sokolov's system is an instance of the Kowalevski type systems and to apply integration procedure developed for those systems, we just need to rewrite relation (12) in the form of (6) and to relate it with corresponding discriminantly separable polynomial (3). Denote by C a biquadratic polynomial such that…”

“…Another novelty of our approach is that Kowalevski's integration procedure from [12] may easily be generalized and applied on a whole new class of systems we introduced in [3], [5]. We called those systems the Kowalevski type systems and basically they are obtained by replacing so called the Kowalevski fundamental equation by an arbitrary discriminantly separable polynomial of degree two in each of three variables.Further steps of integration procedure for those systems follow Kowalevski's procedure and finally we can get explicit solution for all systems of the Kowalevski type in theta functions of genus two.…”