Derivation of the Euler-Lagrange-Equation¶
We would like to find a condition for the Lagrange function , so that its integral, the action , becomes maximal or minimal.
For that, we change the coordinate by a little variation , although infinitesimal. Additionally, has to hold. The integral of the Lagrange function becomes:
This should be extremal with respect to . So we need to differentiate with respect to that and set equal to :
For this total derivative, the partial derivatives of and and have to be found.
For the second summand, we use partial integration:
The middle term is equal to since vanished on the boundary points. Therefore, the last term remains.
Now we can factor out that . The integral vanished for all variations iff the parentheses vanishes.
We yield the Euler-Lagrange-Equation: