Definition:Calculus of Variations

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Calculus of variations is the subfield of analysis concerned maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.

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Historical Note

Some sources suggest that, in a sense, the earliest problem in the calculus of variations arose in one of the legends of the founding of Carthage; the city was granted as much land as could be enclosed by a given length.

The calculus of variations emerged as a branch of mathematics as a result of investigations into the cycloid in the $18$th century.

The first systematic investigation of the topic was given by Joseph Louis Lagrange in his earliest and most important works, together with Leonhard Paul Euler, who coined the term in $1766$.

Karl Weierstrass ushered in a new era of precise reasoning with his lectures in $1879$ on the subject. One of his students, Oskar Bolza, took on the subject and developed the Chicago school of the calculus of variations.