Calc of variations, minimizing functionals question

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Discussion Overview

The discussion revolves around the minimization of functionals in the context of calculus of variations, specifically addressing the treatment of constraints in optimization problems. Participants explore the differences in approaches to including Lagrange multipliers and the implications of various formulations in the context of a physics problem related to Bose-Einstein condensates.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a functional ##A## that needs to be minimized under the constraint ##B[F]=\int (|F|²)=N##, questioning the omission of the ##-µN## term in a physics book's approach.
  • Another participant suggests that since ##N## is an arbitrary constant, it may be acceptable to set ##N=0## for simplification.
  • A different participant clarifies that in the context of the physics book, ##N## represents the number of particles, which complicates the use of the arbitrary constant argument.
  • Participants reference external sources that either include or exclude the ##µN## term in their discussions of integral constraint minimization, highlighting inconsistencies in approaches.
  • One participant questions the equivalence of two formulations of the optimization problem, suggesting that minimizing with respect to ##C[F]## should yield the same results as the original formulation.
  • Another participant agrees with the previous point but notes that neglecting ##µN## can lead to solutions that may not satisfy the constraint if other constraints are present.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of the ##µN## term in the minimization process, with some supporting its exclusion under certain conditions while others argue for its necessity. The discussion remains unresolved regarding the implications of these different approaches.

Contextual Notes

There are limitations regarding the assumptions made about the nature of ##N## and the potential impact of additional constraints on the optimization problem. The discussion does not resolve how these factors influence the validity of different minimization strategies.

Coffee_
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Consider the following problem:

##A## is a functional (some integral operator to be more specific) of a (complex) function ##F##.

We want to minimize ##A[F]## wrt. to a constraint ##B[F]=\int (|F|²)=N##

If I read around online I find that in general such extremization problems are done by minimizing:

##A[F]-µ(B[F]-N) ##

Where ##µ## is a Lagrange multiplier.

My physics book does it slightly differently however, it doesn't include the ##-µN## term in the minimization.

They just say that it's alright to minimize ##A[F]-µB[F]## for a fixed ##µ##.

Why can they use a slightly different way or extremizing a functional under a constraint than the general method I seem to find online?
 
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N is an arbitrary constant, right?
So it should be acceptable to use N=0.
 
No, this is a physics book on Bose Einstein condensates and N is the number of particles in the system. I am seriously confused now because now I've started looking more online for general explanations of this integral constraint minimization and some DO include the ''µN'' part and some don't.

Example where they don't : http://www.mpri.lsu.edu/textbook/Chapter8-b.htm (see section integral constraints)

Example where they DO: http://liberzon.csl.illinois.edu/teaching/cvoc/node38.html (see eq 2.50)
 
Why isn't ##A[F]-µC[F] = A[F]-µ(B[F]-N) ## with ##C = B[F] - N## the same optimization problem?
 
fresh_42 said:
Why isn't ##A[F]-µC[F] = A[F]-µ(B[F]-N) ## with ##C = B[F] - N## the same optimization problem?

It would be if you indeed correctly switch to ##C[F]## which they don't, they explicitly do minimize ##A[F]-µB[F]## but I think I understand it now.

What they do is do not vary µ during the minimization such that ##µN## is just a constant which can be neglected. They will then find a solution ##F(µ,x)## which would need to be plugged into ##B[F(x,µ)]=N## to fix ##µ##.

The difference here is that if you'd minimize ##A[F]-µ(B[F]-N)## wrt. to both ##µ## and ##F## you'd get the correct ##µ## and ##F## simultaneously. Am I correct?
 
As far as I can see (draw some lines ...) you are right as long as there are no other constraints. If there were, the minimal ##A[F]## could have led you to a solution ##F(μ,x)## where the "plug-in" ##B[F(μ,x)] = N## is already out of the allowed area. This cannot happen if you couple it beforehand.
 

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