I was playing around, and I got stuck trying to take the variation with respect to a given Lagrangian.
$L =(1/2)\partial_{u}\theta\cdot\partial^{u}\theta+(\lambda/2)(1-\theta\cdot\theta)$
where $\theta$ is an N-component vector.
We should end up with
$$0 = \partial_{u}\cdot\partial^{u}\theta+ \lambda\theta$$
Could anyone help out and explain this?
Lagrangian: $L =(1/2)\partial_{u}\theta\cdot\partial^{u}\theta+(\lambda/2)(1-\theta\cdot\theta)$
Euler-Lagrange equation: $$ 0 = \frac{\partial}{\partial u} \cdot \left(\frac{\partial L}{\partial(\partial_u\theta)}\right) - \frac{\partial L}{\partial\theta} = \frac{\partial}{\partial u}\left(\partial_u\theta\right) - (-\lambda\theta) = \partial_u \cdot \partial_u\theta + \lambda\theta $$
The calculation is somewhat simplified, but it's in fact not much more than this.