May 2022
Intermediate to advanced
487 pages
9h 22m
English
Content preview from Differential Equations, 3rd Edition
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9The Calculus of Variations
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:
Points P and Q are fixed in the plane. Find the ...
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