Stable solutions using the Euler approximation

A minor modification of the standard Euler approximation for the solution of oscillatory problems in mechanics yields solutions that are stable for arbitrarily large number of iterations, regardless of the size of the iteration interval. The period of a nonlinear oscillator converges rapidly to its exact value as the size of the iteration interval is decreased. In two dimensions, closed orbits are given for the two-body Kepler problem and the restricted three-body problem can be iterated indefinitely to produce space-filling orbits. In this new approximation, the difference ΔE between the initial energy and the energy after n iterations is bounded, oscillatory, and zero when averaged over half a cycle of the motion.