Classical r-matrix structures of integrable mappings related to the Volterra lattice

Abstract

Lax representations and corresponding r-matrices for a Neumann type and a Bargmann type mappings that are generated through nonlinearization of Lax pair from Volterra lattice are presented, and thus, invariants of mappings are obtained and the integrability is established. In addition, the corresponding continuous Hamiltonian systems of invariants of these two mappings are investigated. In the case of Neumann type mapping, the corresponding continuous Hamiltonian flows are shown to be a kind of realizations of the Gaudin magnet with boundary.