On the r-Matrix Structure of the Neumann System and its Discretizations

Abstract

A novel impetus to the construction of integrable discretisations of given integrable continuous-time hamiltonian systems has been given in recent years by a number of relevant findings: we mention the successful application to differential-difference integrable hierarchies of the stationary flow or restricted flow approach [1], the results obtained toward the identification of integrable mappings of the standard type [2], the discovery of Backlund transformations for Calogero-Moser and Rujsenaars systems [3], including relativistic Toda [4], and finally the construction of non-autonomous mappings which are endowed with a proper discrete analog of the Painleve’ property [5]. Of course, time-discretisation is a highly non-unique procedure, even if we restrict considerations to integrability-preserving difference schemes. One may just ask to get an integrable Poisson map such that the discrete dynamics it describes goes into the continuous one in a suitable asymptotic limit, together with integrals of motion and Poisson structure, or require that Poisson structure and integrals of motion be exactly preserved by the discretisation. Stationary or restricted flow technique typically lead to discretisation of the former type, while Backlund transformations provide an example of the latter one. In the present paper, we compare two integrable discretisations of the Neumann system, both belonging to the first family; indeed, one of them is obtained by applying the restricted flow technique to the Toda lattice hierarchy.