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. [1] and Q. [2], 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:

$$

\mathbb{H}=\frac{1}{2}\int^t_0 \left(p(\tau)p(t-\tau)+q(\tau)q(t-\tau)\right)\,\text{d}\tau

\tag{1}$$

This Hamiltonian has the associated Hamilton's equations of (as per Q. [2]) :

$$

\dot{q}(\tau)=p(\tau),\,\dot{p}(\tau)=q(\tau)

\tag{2}$$

How would one formulate a symplectic (or variational) integrator for such a Hamiltonian?

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[1] This question deals with the Legendre transform for non-local Lagrangian formulations.

[2] This question (and answer) deals the derivation of the Euler-Lagrange equations and Hamilton's equations for non-local Lagrangians.