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What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a particular surface which minimize the functional

$$H[y(s)] = \int_0^L ds~\kappa(s)^2 $$

i.e. the total squared curvature along the curve, with vanishing curvature at the end points. I can find solutions numerically very easily, but not the ones I'm interested in.

I've tried some shooting methods, linearizing the ODE about various points, and some gradient flow techniques, all to no avail. Someone much more capable than I am is working on it perturbatively. To be clear, I'm not asking anyone do attempt to do this for me, I'm just asking for some methods of solving this type of problem, or useful books to turn to. Any suggestions?

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You could try to use gradient descent for your functional, with respect to several metrics on the space of curves, $L^2$ with respect to arc length for the metric on your surface, or some higher Sobolev metric. See my homepage for papers on these metrics. – Peter Michor Jul 10 '13 at 15:54
Are you solving the associated Euler-Lagrange equation as two point boundary value problem ? If so, there is huge literature on this. – Piyush Grover Jul 10 '13 at 16:52
I am solving the Euler-Lagrange equation. I'll look into both of your recommendations. Thanks a lot! – ZachMcDargh Jul 10 '13 at 17:26

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