Find the path that minimizes the arclength of the curve between and .
The goal is to minimize arclength as goes from 0 to 1. This quantity is described in this integral:
The Euler-Lagrange equation is defined as:
where is the integrand in the integral above.
Compute the derivatives for the given problem.
since is independent of .
Simplify the expression by factoring out and putting the common denominator on the bottom.
This equation will hold only if is constant so that .
In that case,
and since ,
and so that