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Applying the Euler-Lagrange equations, a special case

  1. Nov 10, 2012 #1
    In the proof to Theorem 7.3 from this paper on FNTFs, the authors invoke the so-called "Langrange equations." I assume they mean the Euler-Lagrange equations. (But maybe not...?) Unfortunately I'm not at all familiar with the Euler-Lagrange equations, and in reading what they are, I have no idea how to apply them in this case.

    If anyone has some spare time and good will, can he/she please explain how to understand this?

    1. The problem statement, all variables and given/known data

    Let [tex]K=\mathbb{C}[/tex] be the complex numbers and [tex]S(K^d)[/tex] the unit sphere in [tex]K^d[/tex] for some positive integer d. Let [tex]\{x_n\}_{n=1}^N\subseteq S(K^d)[/tex] be a fixed sequence in that unit sphere. Let [tex]S=\{(a,b)\in\mathbb{R}^d\times\mathbb{R}^d:\lvert a\rvert^2+\lvert b\rvert^2=1\}[/tex] be the unit sphere in [tex]\mathbb{R}^d\times\mathbb{R}^d[/tex], and define the function [tex]\widetilde{FP}_l:S\to[0,\infty)[/tex] by

    [tex](a,b)\mapsto 2\sum_{n\neq l}(\langle a,a_n\rangle+\langle b,b_n\rangle)^2+(\langle b,a_n\rangle-\langle a,b_n\rangle)^2+1+\sum_{m\neq l}\sum_{n\neq l}|\langle x_m,x_n\rangle|^2,[/tex]

    where the sums are otherwise over 1 through N, and l is some integer between 1 and N. Let [tex](a_l,b_l)\in S\subset\mathbb{R}^d\times\mathbb{R}^d[/tex] be a local minimizer of [tex]\widetilde{FP}_l[/tex].

    Show that there exists a scalar [tex]c\in\mathbb{R}[/tex] such that both of the following equations hold:

    (7.1) [tex]\nabla_a\widetilde{FP}_l(a,b)|_{(a,b)=(a_l,b_l)}=c\nabla_a(\lvert a\rvert^2+\lvert b\rvert^2)|_{(a,b)=(a_l,b_l)};[/tex]

    (7.2) [tex]\nabla_b\widetilde{FP}_l(a,b)|_{(a,b)=(a_l,b_l)}=c\nabla_b(\lvert a\rvert^2+\lvert b\rvert^2)|_{(a,b)=(a_l,b_l)}.[/tex]


    2. Relevant equations

    The "Langrange equations," which I assume refers to the Euler-Lagrange equations.

    Also, I do not understand what the symbols [tex]\nabla_a,\nabla_b[/tex] mean. I would expect they refer to some kind of gradient. But what's with the subscripts? I'm sorry to say I'm more than a little lost.

    3. The attempt at a solution

    I understand most of the rest of the proof to Theorem 7.3. But I just don't know how to interpret this business of Euler-Langrange equations.

    If possible, I would like someone to show me in a textbook (I can get almost anything online or from my university library) what theorem to use, and what choices to make in applying the theorem. For instance, if a theorem calls for a function f, then what is a suitable choice of f in this case?

    Thanks guys.
     
  2. jcsd
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