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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?

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Backward error analysis comes to mind. For example, for the Verlet scheme, you can say that the numerical solution turns out to be the exact solution for a perturbed Hamiltonian system (also known as the shadow Hamiltonian, the resulting system of differential equations are called the modified equations). See also:

S. Reich, “Backward error analysis for numerical integrators,” SIAM Journal on Numerical Analysis, vol. 36, no. 5, pp. 1549–1570, 1999.

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  • $\begingroup$ Thank you for the response! Does this mean that symplectic integrators like the one described in the paper can conserve energy exponentially, but merely volume-preserving algorithms have a worse bound on energy fluctuation? $\endgroup$
    – BRSTCohomology
    Jun 29, 2018 at 20:41

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