Geometric Mechanics and Control Seminar

A new approach to separation of variables in some algebraic integrable systems

Speaker:  Yuri Fedorov (UPC)
Date:  Friday, 16 April 2021 - 15:30
Place:  Online - (ID: 975 2374 3142; Access Code: 044339)

In the first part of the talk I will give a brief survey on different concepts of separation of variables for finite-dimentional integrable systems and describe two new approaches to the separation, based on differential geometric (Kowalewski separability conditions in the theory of Poisson pencils and bi-Hamiltonian geometry) and algebraic geometric (parameterization of Abelian varieties) techniques. In the rest of the talk it will be described how the first approach has been recently applied to obtain a completely new separation of variables for the famous Clebsch integrable case of the Kirchoff equations. For the general case of motion, this system was first reduced to quadratures by F. K"otter in 1891. The quadratures represent a standard Abel map associated to a genus 2 curve. In the new separation procedure led to quadratures which involve points on two different non-hyperelliptic genus 3 curves C,K. The curves have different periods, which means that, in general, the Abel-type map associated with the quadratures is not well defined, therefore cannot be inverted in terms of meromorphic functions of the complex time variable. Following it will show that the periods of the two genus 3 curves are in fact commensurable, and therefore we deal with a well-defined Abel map of a completely new type: it sends the product K x C to the Prym variety of one of the curves, and can be inverted to give a complete meromorphic solution to the Clebsch system. The approach is equally applicable to some other integrable systems, for which a separation of variables was previously unknown.