Disclaimer: this is a wholesale copy of a question I asked on a different site (Physics.SE). I'll be linking to some posts I made on that site here as well, for the sake of brevity (and since they don't directly relate to the question at hand). I read the posting rules here just to make sure I'm not doing anything wrong, I don't think I am. If yes, let me know.
Now, the question:
I recently asked two questions, Q.  and Q. , regarding reformulating non-local Lagrangians as Hamiltonians.
In these questions, the Hamiltonian is formulated as an integral because of it's non-local nature. Additionally, all of the partial derivatives must be replaced by functional derivatives, for the same reason.
My question is, how does one formulate a symplectic integrator for such a Hamiltonian?
In all symplectic integrator derivations I've seen, the Hamiltonian function is used, not the integral. Is there a more generalized approach one can take in this case?
Take for example the case where:
This Hamiltonian has the associated Hamilton's equations of (as per Q. ) :
How would one formulate a symplectic (or variational) integrator for such a Hamiltonian?
 This question deals with the Legendre transform for non-local Lagrangian formulations.
 This question (and answer) deals the derivation of the Euler-Lagrange equations and Hamilton's equations for non-local Lagrangians.