Overview

The problem of variational calculus is to find the form of the function y( x) which makes the integral

(A.1)

an extremum (maximum or minimum), where F is a given function of x, y, and y'.

Assume that y( x) does give the required extremum, and construct a curve y( x) + ?( x) which is very near to y( x) and satisfies the condition ?( x ₁) = 0 = ?( x ₂). For the new curve our integral modifies to

(A.2)

Now making use of the fact that ? and ?' are small, we may expand the integral to the first order, leading to

(A.3)

Integrating the third term by parts (and dropping the parentheses showing the dependence on the various variables) we get

(A.4)

which substituted back into eqn (A.3) gives

(A.5)

We may argue now that y( x) will make the integral an extremum, if the integral remains unchanged for small variations of ?. This will occur when the coefficient of ? vanishes, that is, when

(A.6)

The above equation is known as the Euler differential equation, the solution of which gives the required function y( x).

When F depends on another function z( x) as well, an entirely analogous derivation gives one more differential equation in the form

(A.7)

These two differential equations, (A.6)...