Euler-Lagrange differential equation (elementary)
Let be a twice differentiable function from to and let be a twice differentiable function from to . Let denote .
Define the functional as follows:
Suppose we regard the function and the limits of integratiuon and as fixed and allow to vary. Then we could ask for which functions (if any) this integral attains an extremal (minimum or maximum) value. (Note: especially in Physics literature, the function is known as the Lagrangian.)
Suppose that a differentiable function is an extremum of . Then, for every differentiable function such that , the function , defined as
will have an extremum at . If this function is differentiable, then when .
By studying the condition (see the addendum to this entry for details), one sees that, if a function is to be an extremum of the integral , then must satisfy the following equation:
This equation is known as the Euler–Lagrange differential equation or the Euler-Lagrange condition. A few comments on notation might be in . The notations and denote the partial derivatives of the function with respect to its second and third arguments, respectively. The notation means that one is to first make the argument a function of by replacing the second argument with and the third argument with and secondly, differentiate the resulting function with respect to . Using the chain rule, the Euler-Lagrange equation can be written as follows:
This equation plays an important role in the calculus of variations. In using this equation, it must be remembered that it is only a necessary condition and, hence, given a solution of this equation, one cannot to the conclusion that this solution is a local extremum of the functional . More work is needed to determine whether the solution of the Euler-Lagrange equation is an extremum of the integral or not.
In the special case , the Euler-Lagrange equation can be replaced by the Beltrami identity.
|Title||Euler-Lagrange differential equation (elementary)|
|Date of creation||2013-03-22 12:21:49|
|Last modified on||2013-03-22 12:21:49|
|Last modified by||rspuzio (6075)|
|Defines||Euler-Lagrange differential equation|