I'm not sure we have enough information, but it looks like some kind of conservation law as the divergence of the vector field is zero. So it conserves phase space volume, like in a Hamiltonian system. You probably need a "geometric integrator," and perhaps the Störmer-Verlet method is a good starting point to address your problem.

That method is easily implemented. If we simplify notation a little and consider the system

h' = f(q)

q' = g(h)

which seems to capture the essence of your problem, then the method proceeds as follows. Choose a fixed time step dt > 0, and compute

hmid = h + dt*f(q)/2

qnew = q + dt*hmid

hnew = hmid + dt*f(qnew)/2

Then loop over these three instructions for as many steps as you need to complete the integration. 

So, in summary, what this method does is to take half a step on the first h-equation, then a full step on the q-equation, using the intermediate h-value you just computed, and finally, a second half step on the h-equation, using the updated q value you just computed. This completes a full step of size dt for the entire system.

This method, known as the Störmer-Verlet method will also preserve phase volume. 

Dear Gustaf Söderlind, thank you very much...

You have decorated your variables with a lot of indices which probably are important to you, but which are only distracting to outsiders. It would have been much more informative if you had indicated which quantities are time dependent, and which are just plain constants. As already said by Gustaf, this looks like the Hamilton equations.

For a Hamiltonian of the form

H = a p2 + b ln(q) + f(t) p + g(t) q, a and b constants

There is no way to integrate this equation analytically; the numerical method suggested by Gustaf seems reasonable. However, for a first attempt you may just as well use a standard library routine (which would be of higher order in the time-step than Störmer-Verlet). If f(t) and g(t) should happen to be constant, you can reduce the problem to one first order equation, making use of H being a constant of motion. But it is not obvious to me that this would be of much help, if any.