Is there a smooth solution to minimize this: $$ \int_0^1{x \over {1+k^2f'(x)^2}}dx, f(0)=1, f(1)=0, f'(x)\leq 0, k^2>0. $$ I could "solve" it using a numeric approximation (my algorithm converged so apparently there is a valid local minimum in the function space) but would love to see an analytic solution (algebraic or not, doesn't matter). Thank you so much in advance. Fingers crossed.
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$\begingroup$ I can't tell what the problem is. If you let $k$ get arbitrarily large, your integrals tend to zero. So there would appear to be no minimum. $\endgroup$– Ryan BudneyCommented Apr 21, 2015 at 6:28
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$\begingroup$ @RyanBudney: Maybe $k$ is supposed to be fixed? $\endgroup$– Nate EldredgeCommented Apr 21, 2015 at 6:34
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$\begingroup$ $k^2$ is a constant, a form factor. $\endgroup$– MandrillCommented Apr 21, 2015 at 6:34
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2$\begingroup$ Have you tried Euler-La grange equation? $\endgroup$– Fan ZhengCommented Apr 21, 2015 at 6:40
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$\begingroup$ @ Fan Zheng I did, but it doesn't work (at least not the standard way to use it) because a "zigzag" (going up and down with a small x variation) function would make a very high $f'(x)$ and that would minimize the functional. I don't know how to introduce a monotonic restriction in the Euler-Lagrange method. $\endgroup$– MandrillCommented Apr 21, 2015 at 6:46
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1 Answer
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Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, the profile giving the minimal aerodynamic or hydrodynamic resistance. Here you are assuming axial symmetry, but the problem is even studied under more general assumptions.
So you can find a lot of material from the keywords "Newton problem" and "Calculus of variations". Here is a nice survey paper by Giuseppe Buttazzo (see in particular section 2 and the references therein for the radial case).
answered Apr 21, 2015 at 9:38
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5$\begingroup$ Dude, would you believe that I was trying to find a solution for this equation for more than 20 years (of course I moved on but always revisiting it)? I read about calculus of variations before but only saw the brachistochrone and catenary, I feel stupid for not finding the Newton already solved it but I can't google an equation can I? How would I know it was "Newton problem"? (how did you?) Thank you so much, I really wanted to see the solution before I die (I am not dying, at least not as far as I know, well we are all dying, but you know what I mean), many many thanks!!! $\endgroup$– MandrillCommented Apr 21, 2015 at 10:01
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$\begingroup$ I guess it should be better known than many of the other variational problems. $\endgroup$ Commented Apr 22, 2015 at 3:32