Calculus of variations

This is a beautiful subject
which is sometimes broached
but never (in my experience) given anywhere near its proper due
in elementary classes.

The question is simple:
given some functional on a function space,
find critical points, esp. extrema.

-This- is the context for optimization theory,
not simply finding extrema of polynomials,
and it is also the original context for sup & inf.

It should also feel completely inassailable at first:
this is a hugely infinite-dimensional space:
how can we possibly access this?

No-one seems to start by stating this obvious point:
students seeing this have never optimized even a -linear functional-
over -surface- of -points-, much less a non-linear functional
over an infinite-dimensional function space,
and indeed Newton had no notion of a manifold,
yet solved some difficult geometric questions.

Indeed, this is exactly the point:
the solutions (for the problems that could be solved @ the time,
and for the problems that can be presented to kids) involve a trick,
and people only see the -trick-, not the -context-
(which is intimidatingly vast).

Indeed, how do you even start?

[BTW, I think people minimize -action-, not -length-,
 b/c action is essentially x^2, while length is \abs{x}.]

The key idea is to say:
-Suppose- you have a solution.
What are some properties that it must satisfy?
Ok, now -instead- of looking directly for functions that minimize the functional,
let's look at functions that have this other, easily checked property,
and see if -that- gives us a smaller slate of functions.

For instance, for minizing distance between 2 points,
you must have curvature zero (otherwise you can "cut the corner" and shrink distance; this works for C^2 functions, maybe C^1).


Another technique is to do -abstract- work,
which again reduces you to a smaller slate.

For instance, computing the -gradient- of the functional,
and then use the 1st derivative test!

For "minimizing distance", this (or something like it)
shows that if you move in the direction of curvature,
you decrease distance (indeed, that's what was behind the restriction above),
and thus you must have zero curvature,
i.e., be a straight line.


A systematic way of doing one of these is "Euler-Lagrange" equations,
which generalizes & clarifies the idea used in the brachistochrone:
to solve this (prima facie intractible) variations problem,
you show that a solution must satisfy a PDE
(yeah, that's really helping us out)
which is of a sufficiently special form that we can just use algebra.

More generally, you should think of this as a way to -relate- variations & PDEs, and it can be used to solve either.
The brachistochrone is a great EG of a question that is -hard- in one context (variations), but is -easy- in another context (very special PDE).

[The reason the brachistochrone is easy is b/c the -speed- of the falling bead is given by its -height-, via conservation of energy.

One way to see that it's essentially soluable is:
you really just need to choose between a tradeoff of
"go down to get more speed"
vs
"go directly towards end to use existing speed most efficiently"
...and this is really just a 1-dimensional question,
so it's not surprising that you can solve it
...and then the one catch is finding an equation for the path,
which is basically parametrizing a variety:
classic "plane curves" work.]

-------------------------------------
BTW, I call variantions "geometry"
even though the underlying space is usually a vector or affine space
b/c the -functional- engenders a geometric structure
[Not in the Kleinian sense,
 but in the Riemannian sense.]
