Calculus of variations Calculus of variations deals with functionals (functions of functions), as opposed to ordinary calculus which deals with functions of real numbers. The interest is in extremal functions: those making the functional attain a maximum or minimum value. On of the earliest problems posed in calculus of variations was the brachistochrone brachios = "the shortest", chronos = "time") problem: Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time (click on the image). Newton was challenged to solve the problem in 1696, and did so the very next day... In fact, the solution, which is a segment of a cycloid, was also found by Leibniz, L'Hopital, and the two Bernoullis. Another motivating example for the calculus of variations: Around that time (in the 1960's) Fermat proposed that light travels between two points over the path that is the least time of all the paths (click here for an excellent explanation). Now, why am I boring you with pre 18th century physics? Imagine a monopolist who is in dire need for a profit maximizing price path. Or imagine the government searching for the optimal path of inflation that minimizes the total social loss. Both of them will use variational methods to solve their problems.
Digging deeper: The basic first-order necessary condition in the calculus of variations is the Euler-Lagrange equation . Click here for an excellent introduction to the calculus of variations . (Click | |
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