Lagrangian Optimization Problems Homework

Of course, I endorse Steve Gubkin's answers as manifestly more relevant to multivariable calculus. What follows is a second or third order motivation:

One interesting bit which connects $n$-dimensional constrained analysis to $n+1$ critical point analysis is the following trick: I'll just indicate it for $n=2$: suppose we wish to find extrema of $f(x,y)$ on the bounded constraint curve $g(x,y)=c$.

Define $F(x,y,\lambda) = f(x,y)-\lambda (g(x,y)-c)$

Here $\lambda$ plays the usual role of $z$. In this notation: $\nabla F = \langle \partial_x F, \ \partial_y F, \ \partial_{\lambda} F \rangle$. Then the condition $\nabla F = 0$ yields: $$\nabla F = \langle \partial_x f-\lambda \partial_x g, \ \partial_y f-\lambda \partial_y g, \ g-c \rangle .$$ Which gives us both the colinearity of the two-dimensional gradients of $f$ and $g$ as well as the constraint.

So, one way we can understand the method of Lagrange multipliers is that it is trading a constrained problem in $n$ dimensions for an unconstrained problem in $n+1$ dimensions.

For the start of how this appears in Junior level mechanics see pages 275-281 of John Taylor's Classical Mechanics or the related PSE question. Page 20 or so of Jon-Ivar Skullerud's Notes from MP350 Classical Mechanics has a very readable presentation (quite close to what I saw in school as a physics undergrad). Some nice (mostly math) examples Wikipedia Example page

Finally, to make good on my earlier comment, the Auxiliary variable concept is behind many of the techniques used in modern theoretical physics. At the moment, I can't find a good general article on this. I recall the idea from conversations with other graduate students in physics. Roughly, the idea is when we have a global symmetry of physics then we can implement it as if it was a dynamical symmetry by introducing auxiliary fields. I think the idea is that some of these may have had interactions at an early point in the universe, but, at the moment the symmetry is frozen hence the equations governing the field are merely algebraic. Algebraic symmetries are quite important, the particles seen form representations of these symmetry groups and basically we are able to predict (at times see the $\Omega_-$ story) the existence of new particles from a pattern. All of this said, the use of the term "auxiliary" in physics is much more general than that particular use I sketch above. Generally, there is always a question of whether we leave a constraint unimposed, or apply the constraint. A professor I studied GR from apparently found an (unphysical) solution to quantum gravity after decades of work because he finally decided to work with the unconstrained problem and only apply the constraint at the end of the analysis. It's easy to see your wrong in related rates when you fix everything to constant and get a obviously wrong result. It's much harder to ascertain you are in error when the analysis is based on a partial fixing which leaves some dynamics. Here is a token example of such terminology: BRST and auxiliary field. I think there must be a better article somewhere, but I leave it at this for now.

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