Functional derivatives: Fréchet, Gateaux derivatives and Euler-Lagrange operators

Question

What is the relationship between Fréchet derivatives, Gateaux derivatives and the usual Euler-Lagrange operator (ELO)? Is the ELO the Fréchet derivative, the Gateaux derivative or does it depends on the context and the theory/lagrangian we consider?

Remark: Just as the usual derivative is the rate of change of the function with respect to certain variable, I am waiting for a physical interpretation of the above 3 derivatives and their meaning. For instance, in several variables, the partial derivative is the rate of change of the variable that is being calculated or derived... By the way, some people refers to functional derivatives as Fréchet derivatives and, as some responses said, identify the EL operator as the Gateaux derivative. Is there any subtle point to say that the functional derivate is a Fréchet derivative instead a Gateaux derivative? Are both equivalent to the EL equation? If not in what circumstances are they equivalent and why?

Answer 1

In the derivation of the Euler-Lagrange equations for a functional, one uses the Gateaux derivative/differential of a functional (this functional could be the „Lagrangian action” in point-particle mechanics/field theory).

Here it is denoted by „d”, but let us denote it by $d_G$. Let $\{h_a (\tau)\}_{a=1,...,N}$ be N arbitrary real functions such as $h_a (\tau_1) = 0 = h_a (\tau_2)$. Let $u$ be a real number (called parameter).

Then we take the functional (deriving from the standard 1st order functional of Lagrange): $$I[x_a + u h_a] :=\int_{\tau_1}^{\tau_2} d\tau ~ L\left(x_a (\tau) + u h_{a}(\tau), \dot{x}_a (\tau) + u \dot{h}_a(\tau), t\right) $$

To this we define its Gateau derivative:

$$d_G I[x_a] := \left(\frac{d}{du} I[x_a + u h_a]\right)|_{u=0} $$

One then defines the functional derivative of $I$ as:

$$\begin{equation} d_G I[x_a] =: \int_{\tau_1}^{\tau_2} d\tau ~ h_a (\tau) \frac{\delta I}{\delta x_a (\tau)} \end{equation}$$

If we compute the Gateau derivative and equal it to $0$, we obtain the Euler-Lagrange equations for functionals/actions of 1st order in the derivative.