> It is a difficult problem. Implementing an ODE solver good enough to use for designing spacecraft trajectories is a difficult problem. Implementing an ODE so

> It is a difficult problem. Implementing an ODE solver good enough to use for designing spacecraft trajectories is a difficult problem. Implementing an ODE solver that can handle an n-body problem (with a fairly small n) without leaving user-visible magical parameters is not difficult. Introductory textbooks can explain how to do this. > The reason is that orbital systems with > 2 bodies meet the definition of a chaotic system, a system acutely sensitive to initial conditions. Even if you have a chaotic dynamical system, you can still solve it. You don't solve it in the sense that the trajectory you get is the correct one, because, as you rightly say, that is mathematically impossible for long time periods. But you can solve it in the sense that the solution you get "could" have originated from an initial condition close to the given initial condition. There are many interesting things you can do for complicated ODE problems. But writing ODE solvers of the level of complexity and accuracy we are talking about here is a solved problem.