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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.

critical points, among which are the minima, both local and global, so a similar “func-tional gradient” will distinguish the candidate functions that might be minimizers of the functional. The finite-dimensional calculus leads to a system of algebraic equations for the

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