How is the multidimensional Lagrangian-function partially derived here? Chain-Rule and? I want to understand the following:
The coordinates in our original coordination system are:
$\vec x = (x_1,..., x_{3N})^T,\dot{\vec x} = (\dot x_1,..., \dot x_{3N})^T$
We want to use generalized coordinates:
$\vec q = (q_1,..., q_{3N})^T,\dot{\vec q} = (\dot q_1,..., \dot q_{3N})^T$
The Lagrangian-function is given by $$L=L(\vec x,\dot{\vec x} , t)=L(\vec x(\vec q),\dot{\vec x}(\vec q, \dot{\vec q},t) , t)$$
So $L: \mathbb{R}^{3N}\times \mathbb{R}^{3N}\times \mathbb{R}^+ \rightarrow \mathbb{R}$
I haven't had any multidimensional analysis yet, so it's difficult for me to understand the following partial derivative:
$$\frac{\partial L}{\partial \dot q_j}=\sum_{i=1}^{3N}(\frac{\partial L}{\partial x_i}\frac{\partial x_i}{\partial \dot q_j} + \frac{\partial L}{\partial \dot x_i}\frac{\partial \dot x_i}{\partial \dot q_j}) $$
Can somebody explain to me, what has been done here?
It looked a bit like the chain-rule, so I read this section:
https://en.wikipedia.org/wiki/Chain_rule#General_rule
What I find confusing, is that I can't identify an $f$ and $g$,
if you interpret $\vec x:\mathbb{R}^{3N}\rightarrow \mathbb{R}^{3N}, \vec q\mapsto \vec x(\vec q)$ as a function and $L: \mathbb{R}^{3N}\rightarrow \mathbb{R}, \vec q\mapsto L(\vec x(\vec q))$
$$D(L\circ x)=D_L\cdot D_x\\ =\begin{pmatrix} \frac{\partial L}{\partial x_1}L(\vec x) & ... & \frac{\partial L}{\partial x_{3N} }L(\vec x)\end{pmatrix}\cdot \begin{pmatrix} \frac{\partial x_1}{\partial \dot q_1} & ... & \frac{\partial x_1}{\partial \dot q_{3N} } \\ 
 \vdots &... & \vdots \\  \frac{\partial x_{3N}}{\partial \dot q_1} & ... & \frac{\partial x_{3N}}{\partial \dot q_{3N} }\end{pmatrix}\\ 
=(\sum_{i=1}^{3N} \frac{\partial L}{\partial x_i}L(\vec x)\cdot \frac{\partial x_{i}}{\partial \dot q_1},...,\sum_{i=1}^{3N} \frac{\partial L}{\partial x_i}L(\vec x)\cdot \frac{\partial x_{i}}{\partial \dot q_{3N}}) $$
Analogue for $\dot x$:
$$D(L\circ \dot x)=D_L\cdot D_{\dot x}\\= \begin{pmatrix} \frac{\partial L}{\partial \dot x_1}L(\dot{\vec x}) & ... & \frac{\partial L}{\partial \dot x_{3N} }L(\dot{\vec x})\end{pmatrix}\cdot \begin{pmatrix} \frac{\partial \dot x_1}{\partial \dot q_1} & ... & \frac{\partial \dot x_1}{\partial \dot q_{3N} }\\ 
 \vdots &... & \vdots \\ \frac{\partial \dot x_{3N}}{\partial \dot q_1} & ... & \frac{\partial \dot x_{3N}}{\partial \dot q_{3N} }\end{pmatrix}\\ 
=(\sum_{i=1}^{3N} \frac{\partial L}{\partial \dot x_i}L(\dot{\vec x})\cdot \frac{\partial \dot x_{i}}{\partial \dot q_1},...,\sum_{i=1}^{3N} \frac{\partial L}{\partial \dot x_i}L(\dot{\vec x} )\cdot \frac{\partial \dot x_{i}}{\partial \dot q_{3N}}) $$
So if I add my results, I get ALMOST the given form when comparing the components, the differences are the factors $L(\dot{\vec x}), L(\vec x)$:
$$D(L\circ  x)+D(L\circ \dot x)=(\sum_{i=1}^{3N} \frac{\partial L}{\partial x_i}L(\vec x)\cdot \frac{\partial x_{i}}{\partial \dot q_1},...,\sum_{i=1}^{3N} \frac{\partial L}{\partial x_i}L(\vec x)\cdot \frac{\partial x_{i}}{\partial \dot q_{3N}})+(\sum_{i=1}^{3N}\frac{\partial L}{\partial \dot x_i}L(\dot{\vec x})\cdot \frac{\partial \dot x_{i}}{\partial \dot q_1},...,\sum_{i=1}^{3N} \frac{\partial L}{\partial \dot x_i}L(\dot{\vec x} )\cdot \frac{\partial \dot x_{i}}{\partial \dot q_{3N}}) $$
AND I can't explain why I should add $D(L\circ  x)$and $D(L\circ \dot x)$?
 A: Let us write more generally $\vec x = \vec x(\vec q, \dot{\vec q},t)$. I do not know whether it makes sense physically that we assume a dependency of $\vec x$ from $\dot{\vec q}$ and $t$, but formally it is certainly okay. If $\vec x$ only depends on $\vec q$, then we get $\dfrac{\partial x_i}{\partial \dot{\vec q_j}} = 0$ and $\dfrac{\partial x_i}{\partial t} = 0$ which leads to the formula
$$\frac{\partial L}{\partial \dot q_j}=\sum_{i=1}^{3N}\frac{\partial L}{\partial \dot x_i}\frac{\partial \dot x_i}{\partial \dot q_j} .$$
So let us consider the function
$$\tilde L(\vec q, \dot{\vec q},t) = L(\vec x(\vec q, \dot{\vec q},t),\dot{\vec x}(\vec q, \dot{\vec q},t), t)$$
defined on $\mathbb R^{3N} \times \mathbb R^{3N} \times \mathbb R^+$. Note that I wrote $\tilde L$ to distinguish it from $L$ which is a function of $\vec x, \dot{\vec x},t$. We have
$$\tilde L = L \circ X$$
where $X : \mathbb R^{3N} \times \mathbb R^{3N} \times \mathbb R^+ \to \mathbb R^{3N} \times \mathbb R^{3N} \times \mathbb R^+$ is given by $X(\vec q, \dot{\vec q},t) = (\vec x(\vec q, \dot{\vec q},t),\dot{\vec x}(\vec q, \dot{\vec q},t), t)$.
Since we are only interested in the partial derivatives with respect to the $\dot{\vec q_j}$, we simplify notation by ignoring the variables $\vec q$ and $t$. That is, we consider the functions
$$\vec x = \vec x(\dot{\vec q}) ,$$
$$\dot{\vec x} = \dot{\vec x}(\dot{\vec q}) ,$$
$$L(\vec x,\dot{\vec x}) ,$$
$$\tilde L(\dot{\vec q}) = L(\vec x(\dot{\vec q}),\dot{\vec x}(\dot{\vec q})) ,$$
$$X(\dot{\vec q}) = (\vec x(\dot{\vec q}),\dot{\vec x}(\dot{\vec q})) .$$
The chain rule tells us that the Jacobian matrix of $\tilde L$ at a point $\dot{\vec q} \in \mathbb R^{3N}$
$$J \tilde L (\dot{\vec q}) = \begin{pmatrix} \dfrac{\partial \tilde L}{\partial \dot q_1}(\dot{\vec q}) & \dots  & \dfrac{\partial \tilde L}{\partial \dot q_{3N}}(\dot{\vec q}) \end{pmatrix}$$
is the product $JL(X(\dot{\vec q})) \circ JX(\dot{\vec q})$ of the Jacobians of $L$ and $X$. Suppressing the argument $\dot{\vec q}$ on the RHS we get
$$JL(X(\dot{\vec q})) = \begin{pmatrix} \dfrac{\partial L}{\partial x_1} & \dots & \dfrac{\partial L}{\partial x_{3N}} & \dfrac{\partial L}{\partial \dot x_1} & \dots  & \dfrac{\partial L}{\partial \dot x_{3N}} \end{pmatrix} $$
$$JX(\dot{\vec q}) = \begin{pmatrix}
\dfrac{\partial x_1}{\partial \dot  q_1} & \dots & \dfrac{\partial x_1}{\partial \dot q_{3N}} \\
\dots & \dots & \dots \\
\dfrac{\partial x_{3N}}{\partial \dot  q_1} & \dots & \dfrac{\partial x_{3N}}{\partial \dot q_{3N}} \\
\dfrac{\partial \dot x_1}{\partial \dot  q_1} & \dots & \dfrac{\partial \dot x_1}{\partial \dot q_{3N}} \\
\dots & \dots & \dots \\
\dfrac{\partial \dot x_{3N}}{\partial \dot  q_1} & \dots & \dfrac{\partial \dot x_{3N}}{\partial \dot q_{3N}}
\end{pmatrix}$$
A simple matrix multiplication shows then that
$$\frac{\partial \tilde L}{\partial \dot q_j}=\sum_{i=1}^{3N}(\frac{\partial L}{\partial x_i}\frac{\partial x_i}{\partial \dot q_j} + \frac{\partial L}{\partial \dot x_i}\frac{\partial \dot x_i}{\partial \dot q_j}) .$$
