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calculus of variations |
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| A branch of mathematics concerned with finding functions that minimize or maximize certain quantities associated with them. A classic early problem was the brachistochrone problem: find the curve between two points in a vertical plane along which a particle would move freely under gravity in the shortest possible time. Here, typically, the time taken for a curve y=f(x), with y\' = df(x)/dx is given by an integral of the form ʃ {TAG(tag=>sup)}b{TAG}{TAG(tag=>sub)}a{TAG} F(x, y, y\') dx and the minimizing curve (in this case a cycloid) satisfies Euler\'s equation ∂F/∂y = d/dx(∂F/∂y\'). |
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