how to apply the Langrange equations in this case
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?
Let be the complex numbers and the unit sphere in for some positive integer d. Let be a fixed sequence in that unit sphere. Let be the unit sphere in , and define the function by
where the sums are otherwise over 1 through N, and l is some integer between 1 and N. Let be a local minimizer of .
Show that there exists a scalar such that both of the following equations hold:
I assume that refer to the gradients on , respectively. However I'm not sure about that.
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?