# Symplectic Algorithms for Hamilton’s Equations as opposed to just Volume-Preserving

this might be a silly question, but if we’re trying to numerically solve Hamilton’s equations with some discrete scheme, sometimes when the scheme preserves phase space volume (Hamilton’s eqns are understood to be some flow on this phase space) it leads to stability, even if the algorithm isn’t symplectic (for example the nonrelativistic Boris Pusher algorithm for electrodynamics).

I’m wondering, what is there to be gained in the numerical analysis sense for a discrete, symplectic scheme for Hamilton’s equations? Some of the obvious desiderata like better error bounds might just come from the algorithm more closely resembling the Hamilton flow itself, but in general, can it be said exactly what is gained when an algorithm is symplectic and not just volume-preserving?