Euler-Lagrange Differential Equation

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

where

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

is satisfied.

If time-derivative notation y^. is replaced instead by space-derivative notation y_x, 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, f_x (the partial derivative of f with respect to x) 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 deltaq^.=d(deltaq)/dt. Now, integrate the second term by parts using

so

Combining (◇) and (◇) then gives

But we are varying the path only, not the endpoints, so deltaq(t_1)=deltaq(t_2)=0 and (14) becomes

We are finding the stationary values such that deltaJ=0. These must vanish for any small change deltaq, which gives from (15),

This is the Euler-Lagrange differential equation.

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

where

and the first, second, etc., variations are

The second variation can be re-expressed using

so

But

Now choose lambda such that

and z such that

so that z satisfies

It then follows that

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html

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