Shadowing high-dimensional hamiltonian systems: the gravitational N-body problem.

A shadow is an exact solution to a chaotic system of equations that remains close to a numerically computed solution for a long time. Using a variable-order, variable-time-step integrator, we numerically compute solutions to a gravitational N-body problem in which many particles move and interact in a fixed potential. We then search for shadows of these solutions with the longest possible duration. We find that in "softened" potentials, shadow durations are sufficiently long for significant evolution to occur. However, in unsoftened potentials, shadow durations are typically very short.