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The Calculusof Variations

The calculus of VariationsPeter J. OlverSchool of MathematicsUniversity of MinnesotaMinneapolis, MN olverContents1. Introduction.. 22. Examples of Variational Problems.. 2 Minimal Curves, Optics, and Geodesics .. 3 Minimal Surfaces .. 6 Isoperimetric Problems and Constraints .. 83. The Euler Lagrange Equation.. 9 The First Variation .. 9 Curves of Shortest Length Planar Geodesics .. 12 Minimal Surface of Revolution .. 13 The Brachistochrone Problem .. 16 The Fundamental Lemma .. 19A Cautionary Example .. 204. Boundary Conditions.. 22 Natural Boundary Conditions .. 22 Null Lagrangians .. 25 General Boundary Conditions.

The history of the calculus of variations is tightly interwoven with the history of math-ematics, [12]. The field has drawn the attention of a remarkable range of mathematical luminaries, beginning with Newton and Leibniz, then initiated as a subject in its own right by the Bernoulli brothers Jakob and Johann. The first major developments ...

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