Linearity of variational derivative?

Question

Just wondering. Suppose we have a functional of the form

$$ E(y) = \int_X\mathcal{L}(x,y,y')dx $$

The variational derivative is given by

$$ \frac{dE}{dy} = \frac{d\mathcal{L}}{dy} - \frac{d}{dx}\frac{d\mathcal{L}}{dy'} $$

Because the meaning of the variational derivative is somehow a gradient is there a way to define a linear mapping that maybe I'm not seeing?

I'm asking because usually these expressions are quite complicated and I really can't see any linearity at all.

Answer 1

The variational derivative of the functional $E(y)$ is a functional of two functions, $$ \mathcal{D}E(y)(\phi) = \left. \frac{d}{d\lambda} E(y+\lambda\phi) \right|_{\lambda=0} $$

In general the functional $\phi \mapsto \mathcal{D}E(y)(\phi)$ is not linear, but for example when $E(y) = \int \mathcal{L}(x, y, y') \, dx$ then it is in fact linear.

Wikipedia articles:

• Functional derivative
• Gâteaux derivative
• Fréchet derivative