9The Calculus of Variations

DOI: 10.1201/9781003214526-9

  • The concept of calculus of variations

  • Variational problems

  • Euler's equation

  • Isoperimetric problems

  • Lagrange multipliers

  • Side conditions, integral

  • Side conditions, finite

9.1 Introductory Remarks

In a maximum-minimum problem from calculus, we seek a point at which a given function assumes a maximum or minimum value. The calculus of variations concerns itself with a much more subtle sort of question: We are given a functional, usually defined by an integral, that assigns to each curve in a large family a numerical value. And we want to find the curve that minimizes the functional. To make the idea concrete, let us consider some examples:

  1. Points P and Q are fixed in the plane. Find the ...

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