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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 .. 275. The Second Variation.. 306. Multi-dimensional Variational Problems.. 34 The First Variation and the Euler Lagrange Equations .. 35 References.. 393/21/211c 2021 Peter J. Olver1. principles form one of the most wide-ranging means of formulating math-ematical models governing the equilibrium configurations of physical systems.

1. Introduction. Minimization principles form one of the most wide-ranging means of formulating math-ematical models governing the equilibrium configurations of physical systems. Moreover, many popular numerical integration schemes such as the powerful finite element method are also founded upon a minimization paradigm.

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