Definition:Calculus of Variations
Definition
The calculus of variations is the subfield of analysis concerned maximizing or minimizing real functionals, which are mappings from a set of functions to the real numbers.
Also see
- Results about calculus of variations can be found here.
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.
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): calculus of variations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variation: 3.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): calculus of variations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variation: 3.