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Methods of Mathematical Physics, Third Edition by Bertha Jeffreys, Harold Jeffreys

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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

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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 ...

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