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Calculus of VariationsCalculus of Variations
Euler's EquationEuler's Equation
  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 non-function-like. 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.
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