# Applying the Euler-Lagrange equations, a special case

1. Nov 10, 2012

### hatsoff

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

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

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

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

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

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

2. Relevant equations

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

Also, I do not understand what the symbols $$\nabla_a,\nabla_b$$ 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.