Poisson Brackets of Mappings Obtained as $(q,−p)$ Reductions of Lattice Equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The $(q,−p)$ reductions are $(p + q)$-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the $(3,−2)$ reductions of the integrable partial difference equations are Liouville integrable in their own right.