Euler-Lagrange Differential Equation

The Euler-Lagrange differential equation is the fundamental equation of calculus of variations. It states that if is defined by an integral of the form

where

then has a stationary value if the Euler-Lagrange differential equation

is satisfied.

If time-derivative notation is replaced instead by space-derivative notation , the equation becomes

The Euler-Lagrange differential equation is implemented as EulerEquations[f, u[x], x] in the Wolfram Language package VariationalMethods` .

In many physical problems, (the partial derivative of with respect to ) turns out to be 0, in which case a manipulation of the Euler-Lagrange differential equation reduces to the greatly simplified and partially integrated form known as the Beltrami identity,

For three independent variables (Arfken 1985, pp. 924-944), the equation generalizes to

Problems in the calculus of variations often can be solved by solution of the appropriate Euler-Lagrange equation.

To derive the Euler-Lagrange differential equation, examine

since . Now, integrate the second term by parts using

so

Combining (◇) and (◇) then gives

But we are varying the path only, not the endpoints, so and (14) becomes

We are finding the stationary values such that . These must vanish for any small change , which gives from (15),

This is the Euler-Lagrange differential equation.

The variation in can also be written in terms of the parameter as

where

and the first, second, etc., variations are

The second variation can be re-expressed using

so

But

Now choose such that

and such that

so that satisfies

It then follows that