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.

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 and if *f* has a local minimum (or maximum) at a point *t*_{0}, then (d*f* / d*t*) (*t*_{0}) = 0.

The calculus of variations ...

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