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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...