# Formulating a symplectic (or variational) integrator for a non-local Hamiltonian

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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.

edited May 27, 2015

I can't seem to get your link to Q. [2] to work, but the link to Q. [1] is OK. Could you confirm that your link to Q. [2] is correct?

@JoshBurby thanks for the catch, I have fixed the second link.

I'm curious where this kind of Lagrangian shows up in what you're studying. Or are you just studying it for fun?

@JoshBurby: I don't know about this one, but nonlocal Lagrangians appear whenever one integrates out quadratic variables from an originally bigger Lagrangian model.

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You may proceed in analogy with the derivation of the variational integrators in a paper by Marsden and West, Discrete mechanics and variational integrators, 2002. The basic recipe (starting from the lagrangian formulation) is outlined in the introduction.

You need to figure out the details by yourself as the problem of variational integration of nonlocal problems hasn't been discussed in the literature so far. This means that you need to work harder, but you'll be able to publish your findings once you arrive at a working algorithm.

answered Apr 1, 2015 by (15,468 points)

This definitely looks like a good place to start, thank you! (And thanks for the help on Physics.SE too!)

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