Continuous limits for the Kac-Van Moerbeke hierarchy and for their restricted flows

Abstract

The continuous limit for the Kac-Van Moerbeke (KvM) hierarchy, for their bi-Hamiltonian formulation, recursion relation and square eigenfunction relation is studied. A new family of integrable symplectic maps (ISM) are reduced from the KvM hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions. Their integrability is deduced from the discrete zero-curvature representation of the KvM hierarchy. It is shown that these ISMs provide maps which approximate many well known integrable mechanical systems (e.g. Neumann, Garnier) embedded into the KdV hierarchy as their restricted flows.