Abstract A method for approximating a large class of Hamiltonian ordinary equations by area-preserving discrete mapping equations is described. The time-steps can be quite large compared to conventional approaches. It is shown that, in general, the mapping equations are implicit, and the conditions under which a unique solution can be found are determined. Under these conditions, Newton's method is rapidly convergent. Numerical examples are studied, and the appropriateness of these methods for use in plasma simulations is explored.