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# If The Shortest Distance Between Two Points Is A Straight Line

6. The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve between two points: $$I(y) = \int_{x_1}^{x_2} \sqrt{1 + (y')^2}\, dx,$$ apply the Euler-Langrange equation, and Bob's your uncle.
9. $\begingroup$ "The shortest distance between two points on the sphere is not a straight line." It's not straight when embedded in a 3D Euclidean space, but on the surface of a sphere those lines are as straight as it gets. $\endgroup$ – Emil Mar 22 '16 at