Shadow mass and the relationship between velocity and momentum in symplectic numerical integration

It is often assumed, when interpreting the discrete trajectory computed by a symplectic numerical integrator of Hamilton’s equations in Cartesian coordinates, that velocity is equal to the momentum divided by the physical mass. However, the “shadow Hamiltonian” which is almost exactly solved by the symplectic integrator will, in general, induce a nonlinear relationship between velocity and momentum. For the (symplectic) momentum- and midpoint-momentum-Verlet algorithms, the “shadow mass” that relates velocity and momentum is momentum independent only for a quadratic potential and, even in this case, differs from the physical mass. Thus, naively assuming the standard velocity-momentum relationship leads to inconsistencies and unnecessarily inaccurate estimates of velocity-dependent quantities. As examples, we calculate the shadow Hamiltonians for the momentum- and midpoint-momentum-Verlet solutions of the multidimensional harmonic oscillator, and show how their velocity-momentum relationships depend on the time step. Of practical importance is the conclusion that, to gain the full advantage of symplecticity, velocities derived from interpolated positions, rather than conventional velocity-Verlet velocities, should be used to compute physical properties.