
 
For now, this is my attempt at doing a long derivation. I don't even think it's
worth reading yet unless you're interested in the guts of my Java code.

 
Find a stationary value (hopefully a maximum or minimum) for a functional
which takes as it's input
any function from
to
and returns a real number.
Here we'll consider functionals in the form:
where is just some function of three variables, the
first two of which happen to depend on the last. The path
which makes the value of stationary
is given by the differential equation known as Euler's Equation:
and the boundary conditions
and

 
Before the derivation, we should know what it means for something to be stationary.
There is a relivant discussion of
Stationary Points for Functions which defines
stationary points as points where the first variation (in this case, the first derivative)
is zero. Here is the sumary:
The plan to find the stationary values of our functional
is to vary
everywhere by a small amount and find
where a small change in leads to
a very small (second order or higher) change in
. Keeping the example of
the Brachistochrone or the
Soap Film in mind will be helpful.
Also note that this has no hope of finding an arbitrary path from
to
, only one that is a function.
Each corresponds to one ,
so it can't double back on itself horizontally or loop around or anything
nonfunctionlike. This freedom is possible, however, with a parametric description of
the path and Calculus of Variations of Multiple Dependent Variables

We wish to find
which makes
stationary.


At the place where is
stationary, any small change in
leads to a very small change in
. We introduce a small
change by adding to an arbitrary
function which is zero
at and
since we are only searching
the space of fixed end points. is a real
number.


Using a Taylor Series for Multivariable Functions,
we expand what's under the integral around


The variation of is written
and is defined as
.
(For now I'll just write instead of
keeping in mind that is really a function that
depends on all of those things.)


Following our way of thinking about stationary points for functions, we say
that is stationary when
the first variation vanishes.


Dividing through by , we
can't make any statement yet about what
should be since is arbitrary
up to it's end points.


It does look like a candidate for
Integration by Parts.


Substituting in the integral for
...


The second integral is easy to evaluate and ends up zero since we imposed
the condition in on
that it vanish at the boundaries:
and


Simplifying in preperation for the grand conclusion, we can see that there are
two factors under the integral.


Since is completly
arbitrary between and
, in order for the integral to be zero, the
second factor must be zero. This is called the Euler Equation, and is what
we were out to show.


