  Modified Hamiltonians are used in the field of geometric numerical
integration to show that symplectic schemes for Hamiltonian systems are
accurate over long times. For nonlinear systems the series defining the
modified Hamiltonian usually diverges. In contrast, this paper constructs and
analyzes explicit examples of nonlinear systems where the modified Hamiltonian
has a closed-form expression and hence converges. These systems arise from the
theory of discrete integrable systems. We present cases of one- and two-degrees
symplectic mappings arising as reductions of nonlinear integrable lattice
equations, for which the modified Hamiltonians can be computed in closed form.
These modified Hamiltonians are also given as power series in the time step by
Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example
displays an implicit dependence on the time step which could be of relevance to
certain implicit schemes in numerical analysis. In the light of these examples,
the potential importance of integrable mappings to the field of geometric
numerical integration is discussed.
