Applying the Euler-Lagrange equations, a special case

[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?

Homework Statement

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]

Homework 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.

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.

 

[mighty2000]

Thank you for your question. The Euler-Lagrange equations are a set of differential equations used to find the extrema (minima or maxima) of a functional. A functional is a function of a function, rather than a function of a variable. In this case, the functional is \widetilde{FP}_l, which is a function of the functions a and b.

The symbols \nabla_a and \nabla_b refer to the gradient with respect to a and b, respectively. The subscripts indicate which variable the gradient is taken with respect to. The gradient is a vector of partial derivatives, which gives the direction and magnitude of the steepest ascent of a function.

To apply the Euler-Lagrange equations in this case, you will need to use the theorem for finding the extrema of a functional. This theorem states that if a function f(x) has a stationary point (a point where the derivative is equal to zero), and if the functional \int_a^b F(x, f(x), f'(x))dx has a stationary point at f(x) = y, then the function f(x) satisfies the Euler-Lagrange equation:

\frac{\partial F}{\partial f} - \frac{d}{dx}\left(\frac{\partial F}{\partial f'}\right) = 0

In this case, the functional is \widetilde{FP}_l, and the stationary point is (a_l, b_l). To apply this theorem, you will need to choose a suitable function F(x, f(x), f'(x)), which will depend on the problem at hand. You will then need to take the partial derivatives with respect to a and b, and set them equal to zero in order to solve for c.

I hope this explanation helps. If you have any further questions, please do not hesitate to ask. Best of luck with your studies.