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Variational Calculus - Proving a functional has no broken extremals?

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi, I am in a variational calculus class and am working on a homework and need a bit of help with two of the problems.

    the first one is to prove that the functional:

    [tex]J(u)= \int (A u'^2 + B u u' + C u^2 + D u' + E u) dx[/tex] where A,B,C,D,E are constants and [tex]A \neq 0[/tex] has no broken extremals.

    The other problem is to show whether or not y'(x)^3 has any broken extremals with the end conditions y(0)=0 and y(x1)=y1


    2. Relevant equations

    I believe we are supposed to use Hilbert's theorem which is stated below:

    Suppose that [tex]F(x,u,p)[/tex] is continuously differentiable for [tex]a \leq x \leq b[/tex] and for all [tex](u,p) \in \mathbb{R}^2[/tex]. Suppose that [tex]F_p[/tex] is continuously differentiable and that [tex]F_{pp} (x,u,p) > 0[/tex] for all [tex]a \leq x\leq b[/tex] and for all [tex](u,p) \in \mathbb{R}^2[/tex]. If u is a weak, piecewise continuously differentiable, local extremal for [tex]J(u)= \int_a^b F(x,u(x),u'(x)) dx[/tex] then [tex]u \in C^2 [a,b][/tex].



    3. The attempt at a solution


    For the first problem, it is evident that all of the conditions for Hilbert's theorem hold except that Fpp = 2A could be negative if A is negative, which seems to not work with this theorem...

    I am thinking that possibly the theorem was incorrectly stated and it is that Fpp<>0 instead of just strictly greater, which would fit the format of the question... unfortunately, I cannot find this theorem stated online or in our text, so i cannot confirm this

    anybody have any idea?


    *** EDIT: for the second question.... i'm an idiot.... i figured it out, now its just the first question...
    For the second question, It seems that Hilbert's theorem most definitely does not hold true, yet since this theorem goes only one way, I was going to try using the Weierstrass-Erdmann conditions along with the euler-lagrange equation to see if I can find an example of a broken extremal....

    I haven't tried this yet and will do so now, but I am not very confident yet that this will work so if there is a different direction that someone could point me in that would be helpful


    anyway, i appreciate any and all help... thanks.
     
    Last edited: Mar 8, 2009
  2. jcsd
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