Euler-Lagrange equation notation: $\delta$ instead of $\partial$

Question

I have seen the equation written as:

$$\frac{\delta L}{\delta q} - \frac{d}{dx} (\frac{\delta L}{\delta \frac{dq}{dx}}) = 0$$

Here, "variation of $L$ divided by variation of $q$ or $\frac{dq}{dx}$" replaces the "corresponding" partial derivatives..

How does this make sense?

Answer 1

When working with more general (topological vector) spaces you extend the notion of (directional) derivative to work in these spaces, so you use another symbol to make it clear you are no longer differentiating a real or complex function (since it's actually a different functional than the usual derivative!).

Depending on which space of functions you're working on you'll have different definitions of directional derivative. When you have a Banach space you can define the Fréchet derivative, and in more general cases the Gâteaux derivative.