An excerpt from an answer:
Consider the Lagrangian:
$L = (0.5)m(\dot{x}^2+\dot{y}^2 +\dot{z}^2)-U$.
The lagrange equation :
$\dfrac{\partial{L}}{\partial{x}}=\dfrac{d}{dt}\dfrac{\partial{L}}{\partial{\dot{x}}}$.
Evaluating this, we get:
$ -\frac{\partial{U}}{\partial{x}} = F_{x} $.
My issue with the evaluation:
The evaluation suggests $\frac{\partial{\dot x}}{\partial{x}}=0$ and $\frac{d}{dt}\left(\frac{\partial{U}}{\partial{\dot x}}\right)=0$, which unfortunately doesn't make sense to me.
How can we say in general that $\dot{x}$ is always not a function of $x$? We might have a situation where, say,$\dot{x} =ax+b$. How do we justify $\frac{\partial{\dot x}}{\partial{x}}=0$ for such a scenario?
If the answer is somehow "treat everything except $x$ as constant", then I will need some justification on what comes under "everything",especially $\dot{x}$, which might have a dependence on $x$ as I point out.
Best Answer
The whole point of lagrangian mechanics is to start rather more abstractly from a configuration space of two variables, without any attached trajectory. At the moment, we can vary $x(t)$ and $x’(t)$ independently, and this independent variation of these quantities just corresponds to different paths between two events. Now Lagrange discovered (and this is where the physics comes in), that if you draw all the possible paths between two events in this configuration space, the one that extremizes the action is the one that nature follows.
So to answer your question: the $x(t)$ and $x’(t)$ are unrelated because they are the configuration space variables first and foremost, and are completely independent. Only when you fix a path do they have any correlation.