
Symplectic Integration
Is anyone here familiar with symplectic numerical integration or Hamiltonian mechanics? I'm looking into numerical integration schemes for orbital mechanics and it seems that symplectic integrators are needed to keep the system from flying apart in long running simulations.
So far, I know that the Verlet scheme is symplectic, but in looking for higher order schemes the only one I found (the ForestRuth algorithm) requires recomputing the forces at three different positions for each time step, which might be a bit much to do real time.
I was looking a little closer at the Verlet scheme though, and I can cancel out the fourth order error term by using the accelerations from previous time steps. What I don't know how to check though is whether or not this modified scheme is still symplectic. (I suspect it isn't or I would think someone else would have mentioned it. That, or maybe there's something else wrong with it.)
The original Verlet scheme is:
The modified scheme is:

I guess no one's familiar enough with symplectic integration, but can anyone familiar with Hamiltonian mechanics and/or tensors help me work through this?
According to Wikipedia, Hamilton's equations are:
Where q is the position and p is the momentum.
I've also got that , and that is the gradient of the potential energy. My first question is how to apply that definition to Newtonian gravity? (T is pretty straightforward, but I'm having trouble working out V.)
Second, it says that the time evolution of Hamilton's equations is a symplectomorphism, meaning it conserves the twoform . As best I can tell, this is a tensor and I'm more or less completely unfamiliar with tensors. (A symplectic integrator is one that also conserves this twoform, which gives it certain good global properties.)