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On some integrable systems related to the Toda lattice. (English) Zbl 0935.37037
Integrable lattices introduced recently by R. Yamilov are shown to be closely related to the usual Toda lattice by means of a sort of Bäcklund transformations. These lattices correspond to three different parametrizations of the phase variables by means of canonically conjugated coordinates and momenta, resulting in three different invariant Poisson structures. The author applies the general procedure of integrable discretization and obtains their integrable finite-difference approximations. Explicit integrable discretizations of these lattices are considered and they are shown to be different tri-Hamiltonian discrete-time systems, whose natural phase space is that of the relativistic Toda lattice.
##### MSC:
 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws 37N20 Dynamical systems in other branches of physics 70H05 Hamilton’s equations 70F99 Dynamics of a system of particles 37K35 Lie-Bäcklund and other transformations