Leapfrog, Cont'd

Time reversibility is important because it guarantees conservation of energy, angular momentum, etc. (in many cases).
- Suppose the integrator makes an error  after one orbital period. Now reverse. Is the error -? No! The time-reversed orbit is a solution of the original ODE (with v replaced with -v), so the energy error is still +. But we've returned to our starting point, so we know the final energy error is zero. Hence  = 0!
Leapfrog is only second order, but very stable.

This argument applies to a 2-body system...

Time reversibility is important because it guarantees conservation of energy, angular momentum, etc. (in many cases).