Calculus of Variations

Area Example

Colin Mitchell

Suppose we want to find the maximum area we can enclose in a non-self intersecting curve of a fixed length L. First, let's consider a parametric definition for the curve, i.e.,

,

and that both of these functions are twice differentiable. We know from calculus then that the area enclosed by the curve is

and the arc length is a constraint of the form

.

We can then assign

.

We now can form a system of Euler equations in t:

Inserting into this

we get

Since these derivatives are equal to zero, the functions being differentiated must be constant, so we find that

Squaring both of these and adding them yields

Clearly, this is a circle.