Abstract
The authors judge symplectic integrators by the accuracy with which they represent the Hamiltonian function. This accuracy is computed, compared and tested for several different methods. They develop new, highly accurate explicit fourth- and fifth-order methods valid when the Hamiltonians is separable with quadratic kinetic energy. For the near-integrable case, they confirm several of their properties expected from KAM theory; convergence of some of the characteristics of chaotic motions are also demonstrated. They point out cases in which long-time stability is intrinsically lost.