Extremal problems were the object of mathematical research at the very earliest stages of the development of mathematics. The first results were then systematized and brought together under the heading of the calculus of variations, with its innumerable applications to physics and mechanics. Attention was devoted principally to the analysis of smooth functions and functionals defined over the entire space or restricted to some smooth manifold. The extremum conditions in this case are the Euler equations (with Lagrange multipliers in the case of constraints).
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