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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.
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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 
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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
non-function-like. This freedom is possible, however, with a parametric description of
the path and Calculus of Variations of Multiple Dependent Variables
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We wish to find
which makes
stationary.
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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.
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Using a Taylor Series for Multivariable Functions,
we expand what's under the integral around
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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.)
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Following our way of thinking about stationary points for functions, we say
that is stationary when
the first variation vanishes.
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Dividing through by , we
can't make any statement yet about what 
should be since  is arbitrary
up to it's end points.
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It does look like a candidate for
Integration by Parts.
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Substituting in the integral for
 ...
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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
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Simplifying in preperation for the grand conclusion, we can see that there are
two factors under the integral.
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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.
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