Alpha Geek
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In relativity theory, velocities can be represented by points in a certain "rapidity space" in which the distance between two neighboring points is 
 where r and ϕ are polar coordinates, and we consider just a two-dimensional space. (An expression like this for the distance in a non-Euclidean space is often called the metric of the space.) Use the Euler—Lagrange equation to show that the shortest distance from the origin to any other point is a straight line.