> does it have conservation of energy? Can you track total energy somehow?

You're going to need a geometric integrator for that, probably an energy-momentum integrator. In practice it is often better to conserve the symplectic form instead of energy, though, and you can show that for fixed time-steps you can't get all of energy/momentum/symplecticity at the same time (I haven't looked at this for a while, a student of the school of Marsden jump in if I'm saying anything wrong). Furthermore, these guarantees all require integrability...

My (ongoing) thesis is in the field of symplectic integrators.

I was asking about these simplified physics engines (in games, etc.) that are probably not deriving physical laws out of whole conservation- (or Noetherian symmetry-) cloth.

Very cool! Game rigid body engines (to my knowledge) usually use symplectic Euler (v^{n+1} = v^n + h M^{-1} F^n, q^{n+1} = q^n + h v^n) with a Moreau/Stewart-Trinkle velocity level discretization of non-penetration constraint (effectively impulses), usually lobbing off the quadratic velocity term (so no precession). These constraints are typically solved in a (projected) Gauss-Seidel fashion to horribly loose tolerances, with liberal post-stabilization thrown in. Outside of rigid bodies, for the most part in games, these days, you don't get 'simulations' that can be derived from a variational principle, but that are instead constraint satisfaction problems with some notion of momentum hacked in the side (see 'Position Based Dynamics', 'Shape Matching').

The movie folks have larger compute time budgets, and I've seen various geometric integrators used there. I know of at least one studio that uses implicit midpoint for thin shell simulation. The Cal-Tech influence has wormed its way into the graphics research world, at least, and you see some papers pop up there frequently looking at geometric integrators. This Danish guy (I'm forgetting his name) at Disney publishes papers every now and then on the topic.

Amazing post!

It is surprisingly difficult to find a concrete explanation of symplecticity. It seems like https://en.m.wikipedia.org/wiki/Symplectic_integrator has links to all the relevant information, but even there the conserved quantity is given as a differential form dp /\ dq, which makes me wonder whether you can even compute it for individual points, or whether you have to integrate over the whole space.