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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.94 Variational Methods

Lawrence C. Evans

The calculus of variations is both a theory in itself and a toolbox of techniques for studying certain kinds of (often extremely nonlinear) ordinary and partial differential equations. These equations, which arise when we seek critical points of appropriate “energy” functionals, are usually far more tractable than other nonlinear problems.

1 Critical Points

Let us begin with a simple observation from first-year calculus, where we learn that if f = f (t) is a smooth function defined on the real line Image and if f has a local minimum (or maximum) at a point t0, then (df / dt) (t0) = 0.

The calculus of variations ...

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