I'm trying to understand eq. (2.2) in the following excerpt taken from this lecture notes (p. 10):
I'm sure that this result is rather trivial, but I've got a hard time to follow the notation. Since I'm not sure in which sense the derivatives have to be understood, I've provided the definition of Gâteaux and Fréchet derivatives I'm familiar with below this question$^1$.
Moreover, let me try to restate the assumptions in the way I understand them. We have a some $L\in C^2(\mathbb R^n\times\mathbb R^n\times\mathbb R)$ (they write $L:\mathbb R^{n+1}\to\mathbb R$ which is clearly a typo), $n\in\mathbb N$, and some $t_0,t_1\in\mathbb R$ with $t_0<t_1$. Now I think they define $$I(\gamma):=\int_{t_0}^{t_1}L(\gamma(t),\gamma'(t),t)\:{\rm d}t\;\;\;\text{for }\gamma\in E:=C^1([t_0,t_1],\mathbb R^d).$$
Now, for $(2.2)$, they take $\gamma,\eta\in E$ (maybe they even assume that $\gamma$ and/or $\eta$ even belong $C^2([t_0,t_1],\mathbb R^d)$) with $\eta(t_0)=\eta(t_1)=0$.
Q1: Am I missing something or isn't $(2.1)$ simply the Gâteaux derivative of $I$ at $\gamma\in E$ in direction $\eta\in E$?
Q2: What they denote by $\frac{\partial L}{\partial q}$ is simply the Fréchet derivative ${\rm D}_1L$ of $L$ with respect to the first argument, isn't it? Analogously, $\frac{\partial L}{\partial\dot q}$ should be simply the Fréchet derivative ${\rm D}_2L$ of $L$ with respect to the second argument.
Q3: If I'm right so far, I really struggle to understand what they denote by $\frac{\rm d}{{\rm d}t}\frac{\partial L}{\partial\dot q}(\gamma(t),\gamma'(t),t)$. If I'm using this notation, I mean the derivative at $t$ of the map $$t\mapsto\frac{\partial L}{\partial\dot q}(\gamma(t),\gamma'(t),t)\tag5,$$ I strongly doubt that this is what they mean, since I don't think their equality would hold then. So, maybe they mean the Fréchet derivative ${\rm D}_3{\rm D}_2L$ of ${\rm D}_2L$ evaluated at $(\gamma(t),\gamma'(t),t)$ instead?
Q4: How can we state and prove $(2.2)$ rigorously?
$^1$ Let $E_i$ be a normed $\mathbb R$-vector space.
1: If $\Omega\subseteq E_1$ is open, $f:\Omega\to E_2$, $x\in\Omega$ and $h\in E_1$, then $f$ is called Gâteaux differentiable at $x$ in direction $h$ if $$t\mapsto f(x+th)\tag1$$ is differentiable at $0$. In that case, $${\rm d}f(x;h):=\left.\frac{\rm d}{{\rm d}t}f(x+th)\right|_{t=0}.$$ $f$ is called Gâteaux differentiable at $x$ if it is Gâteaux differentiable at $x$ in direction $h$ for all $h$.
2: If $\Omega\subseteq E_1$ is open, $f:\Omega\to E_2$ and $x\in\Omega$, then $f$ is called Fréchet differentiable at $x$ if $$\frac{\left\|f(x+h)-f(x)-{\rm D}f(x)h\right\|_{E_2}}{\left\|h\right\|_{E_1}}\xrightarrow{h\to0}0\tag2$$ for some ${\rm D}f(x)\in\mathfrak L(E_1,E_2)$. In that case, $${\rm D}f(x)h={\rm d}f(x;h)\;\;\;\text{for all }h\in E_1\tag3.$$
3: If $\Omega\subseteq E_1\times E_2$ is open, $x\in\Omega$ and $f:\Omega\to E_3$ is Fréchet differentiable at $x$, then $${\rm D}f(x)h={\rm D}_1f(x)h_1+{\rm D}_2f(x)h_2\;\;\;\text{for all }h\in E_1\times E_2\tag4.$$