OK, here's the situation, as I understand it at the moment:
Symplectic integrators, such as the Euler-Cromer method,
are well suited to Hamiltonian dynamical systems.
It turns out that the system I was dealing with last month --
a model of the aeronautical "turns on pylons" maneuver -- is
*not* a Hamiltonian dynamical system. The model does not
conserve energy; not even close. That's because the model
does not account for engine thrust, aerodynamic drag, or
the component of gravity along the aircraft trajectory
during climbs and descents. I know from experience that
these contributions, taken together, have little effect on
the maneuver, so I am justified in leaving them out of the
model ... but it makes the model ineligible to use the
Euler-Cromer method.
It would be nice to have a name for the method I described
this morning, using stride=2 for one integration and second-
order Euler for the other ... but I don't think it would be
appropriate to name it after me. I can't cite any literature
that says to do it exactly that way, but I would be quite
astonished if it was original. I suspect any competent
"old school" numerical analyst would do something like it,
or something better, as a matter of routine. For now I
will just call it the "old school" method.
Bottom line:
a) The Euler-Cromer method is simple and works great
whenever you have a reasonably complete description
of the physics, sufficiently complete that energy is
conserved.
b) The "old school" method is worth keeping in the
toolbox for situations where you need to solve a
differential equation that is not entirely (or not
at all) physics-based.