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I have a variational problem where the unknown function is a periodic path $\gamma:[0,1)\to\mathbb{R}^2$, and the functional is $$ \int_0^1\left( \tfrac12\|\dot\gamma(s)\|^2 + \mathcal{F}[\gamma]\right)\,ds. $$ Here $\mathcal F$ depends on values of $\gamma$ at different values of $s$, so it's not just a function of $\gamma(s)$, like so: $$ \mathcal{F}[\gamma](s) = \sum_{0<j<n}\frac{1}{\|\gamma(s+j/n)-\gamma(s)\|}. $$

My current approach is to expand $\gamma$ into its Fourier components $e^{2\pi \mathrm{i} k s}$ to ensure periodicity, and use a Newton-type method to minimize the functional as a function of the Fourier coefficients. I can calculate the functional and its gradient efficiently for any given set of Fourier coefficients, in part because the integral can be done with the trapezoid rule, which converges quickly for periodic functions.

One of the problems I have is that it is difficult to explore all the different minima the functional might have, because apart from some linear constraints I can impose on the Fourier coefficients, I have very little control over which local minimum I get.

Is there a good graduate/research-level book on numerical variational methods that would explain the different things I should try? Survey articles?

Edit: the exact functional I'm using comes from the n-body choreography problem, but I'm interested more in the general ideas here than in this specific application.

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    $\begingroup$ You might first try looking at the optimal control literature. I don't know of a good textbook off-hand, but that field examines similar types of problems. $\endgroup$ – Geoff Oxberry Jan 13 '15 at 3:29
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    $\begingroup$ It would help if you stated the form of $\mathcal F$. $\endgroup$ – Wolfgang Bangerth Jan 13 '15 at 12:31
  • $\begingroup$ @WolfgangBangerth I put it in the question, but I'm not sure how it helps. It comes from here. $\endgroup$ – Kirill Jan 13 '15 at 21:06
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    $\begingroup$ Ugh, yes, that's an awkward functional. The first part wants to minimize the variation whereas the second wants to maximize it. I'm at least not seeing anything that could be done analytically (but you already knew that). $\endgroup$ – Wolfgang Bangerth Jan 14 '15 at 3:44

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