One of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the "modified equations," which are the equations that are exactly satisfied by the (approximate) discrete solution. These do not actually exist in general, but rather are defined by an asymptotic expansion in powers of the discretization parameter. Nonetheless, if the expansion is suitably truncated, the resulting modified equations have a solution which is remarkably close to the discrete solution. In the case of a Hamiltonian system of ordinary differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. The existence of a modified Hamiltonian is an indicator of the validity of statistical estimates calculated from long time integration of chaotic Hamiltonian systems. Evidence for the existence of a Hamiltonian for a particular calculation is obtained by calculating modified Hami...