## Chapter 10CALCULUS OF VARIATIONS

When change itself can give no more’Tis easy to be true.

SIR CHARLES SEDLEY, *Reasons for Constancy*

### 10·01. Condition for an integral to be stationary.

Suppose that we have an integral of the form

where *f* is a given function; *x* is to be a function of *t*, but we have not yet specified what function. The problem of the calculus of variations is to decide what function *x* must be in order that *S* may be stationary for small variations of *x*. In its simplest form we can consider the determination of the shortest distance between two points. Using Cartesian coordinates and assuming that *y* is a differentiable function of ...