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        riddles >> putnam exam (pure math) >> Calculus of variations
        
(Message started by: Sameer on Oct 6^th, 2007, 2:25pm)

Title: Calculus of variations
Post by Sameer on Oct 6^th, 2007, 2:25pm

Find the plane curve of fixed perimeter and maximum area.

Title: Re: Calculus of variations
Post by towr on Oct 6^th, 2007, 2:53pm

The answer is easy enough, of course. A curve doesn't have an area ;)
[hide]And the curve that encloses the largest area, given the length of the perimeter, is of course a circle.[/hide]
But I still (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?action=display;board=riddles_putnam;num=1165701864;start=0#14) can't remember how to solve it; I'd have to consult my class notes (and considering how they're written, that might not help much either).

Title: Re: Calculus of variations
Post by Sameer on Oct 6^th, 2007, 2:59pm

on 10/06/07 at 14:53:39, towr wrote:

Cool, I forgot I had posted that problem from my book earlier. Yea well the answer is easy, proving is the challenge!! :)

Title: Re: Calculus of variations
Post by Pietro K.C. on Oct 17^th, 2007, 1:36pm

Quote:

A curve doesn't have an area

Watch out! :) I myself tried to prove this for a couple of days, only to find out that, amazingly, some curves do have an area. Indeed, it is possible for a curve to fill out an entire square! The technical statement is that there is a continuous

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif: [0,1] --> [0,1] x [0,1]

which is surjective (but it is never injective). Though we may at first be surprised that we cannot map the "thin" space [0,1] into the "fat" space [0,1]² without repeating values --- which intuitively only "spends" our already very "scarce" domain set --- the fact is that the injectivity restriction is much direr, because the curve then has to dodge itself in ingenious ways to get to places it hasn't been yet.

So ingenious, in fact, that it's impossible.

In the simpler constructions, like the Peano curve, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif is very strongly not injective, but, as suggested by Hilbert and proved by Polya, it is possible to do it with non-injectivity degree 3: every point of the square having at most 3 preimages. The book "Space-Filling Curves" by Hans Sagan gives quite a nice tour of the subject.

The Wikipedia article (http://en.wikipedia.org/wiki/Space-filling_curve)

The proof of non-injectivity of a square-filling curve is actually quite simple: if X is a compact topological space and f : X-->Y a bijective continuous function, then f is actually a homeomorphism; and any number of topological properties distinguish [0,1] from [0,1]² (e.g. it is possible to disconnect [0,1], but not [0,1]², by removing one point).

A natural question might be: is there a curve which has both a perimeter and positive area?

Title: Re: Calculus of variations
Post by towr on Oct 17^th, 2007, 3:04pm

I actually knew about the Hilbert curve; it came up in one of my classes once..
I think it's safe to say any curve of finite length lacks an area, though.

Title: Re: Calculus of variations
Post by Pietro K.C. on Oct 17^th, 2007, 5:11pm

About the original question, I always thought it should be doable by plain symmetry considerations. If S¹ is the unit circle and

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif : S¹ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif²

is continuous and injective, by the Jordan curve theorem it bounds a region R, and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(S¹) is its boundary, denoted http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/partial.gifR. If we suppose for a moment that http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/partial.gifR has zero measure, then R is measurable (has a well-defined area), and the function

g(a) = area of the part of R that lies to the left of the line x=a

is continuous. Since, for large negative a, g(a) = 0, and for large positive a, g(a) = area of R, there must be some place somewhere in between where the line x=a exactly bisects the area of R. We therefore choose the half with smaller perimeter and mirror it across x=a, obtaining a curve which has at most the same perimeter, same area, and is symmetric about x=a.

Of course, all this works for lines in any direction, not just vertical ones. Enumerate the rational numbers in a sequence as (q_n)_{nhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/epsilon.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif}, and define a sequence of regions

R₀ = R
R_n+1 = F(q_n,R_n)

where F(q_n,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/spot.gif) is the process just indicated, but using a symmetry line that makes an angle q_n with the x-axis.

Supposing R to be convex (a reasonable assumption for a maximizing curve, but which I can't prove off the top of my head), each F(q_n,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/spot.gif) acts naturally on the curve's parametrization function http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif, turning it into one which describes the boundary of the new region. With some work, I think one can then prove that, with copious abuse of notation,

F(q_n,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif circle as n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif.

Any thoughts?

Title: Re: Calculus of variations
Post by Sameer on Oct 17^th, 2007, 7:03pm

on 10/17/07 at 17:11:41, Pietro K.C. wrote:

Any thoughts?

You can try a simpler approach using Euler's formula.

Title: Re: Calculus of variations
Post by Pietro K.C. on Oct 17^th, 2007, 9:57pm

Quote:

You can try a simpler approach using Euler's formula

Is that the one in Euler's theorem? ::) (http://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler)

I'm kidding. :) But this solution by symmetry is one of those things which I just haven't sat down to work out fully, even though I have a hunch it might work.

One nice feature of the method is that it gives you existence of a maximum area together with one maximizing curve. I think some finer analysis of the symmetrization might even yield uniqueness.