Forgive me my maybe-non-understanding questions, but this topic seemed quite hard to grasp and the only choice I think I got is ask it as if what I think I believe and you can point me where I'm wrong. The first 4 questions are not really the questions, but my assumptions and just need to know if I'm right about them. The question 5 is the real question. It's all based on this answer
In the variational calculus, we have: \begin{eqnarray} L = \int f(t, x, x') dt \end{eqnarray}
Question 1: In the integral, at first, $x$ and $\dot x$ are really functions of $t$, but we don't treat $\dot x$ to be the derivative of $x$ yet, which means, we leave the possibility that $x(t)$ might be $t^2 + 5t$, where as $\dot x(t)$ might be $t^4 + 15t$ (you can see that one is not derivative of other). At least this is how I think it's in Lagrangian mechanics. This logic comes from the fact that before arriving at true path, they can be independent. It's like we pick a point $t=1$(considering boundaries are from $t=1$ to $t=2$), then we pick a random number and put it instead of $x(t)$, then we pick a random number again and put it instead of $\dot x(t)$ in the integral, we detect what we got(remember it) and do the same thing for $t=1.0000...1$(infinitely close). So at each point of time, we choose 2 random numbers for $x(t), \dot x(t)$ and in the end, get the summed up value. Then we do everything all over again and choose 2 different random numbers at $t=1$ and get to the end and then in the end, we compare all the summed up values and choose minimum. I wanted to explain it like this because I'm wondering if this intuition is correct ? The reason is I want to know if I understand that $x(t)$ and $\dot x(t)$ are independent at first in the integral. Thoughts ?
Question 2: I think even better way of explaining this would be in order for something to be minimal between 2 points(ex.$t=1$ to $t=3$), then it must be minimum between infinetisemal interval. Imagine object starts moving at $t=1$. Then we pick a time($t=1 + dt$), and at $1 + dt$, we pick 2 random numbers for $x(t)$ and $\dot x(t)$ and what should actually happen is even on this interval, $f(t, x, x')$ must be minimum, because if it's not, in the end, whatever we end up with the whole integral, it wouldn't be a minimum. So it's like if you got $\int_{1}^{3} \frac{1}{2}m\dot x(t)^2 - mgx(t)$, we plug in 2 random numbers for $x(t), \dot x(t)$ and $\frac{1}{2}m\dot x(t)^2 - mgx(t)$ must be minimum for these 2 numbers, if not, there will definitely exist other 2 numbers.
Question 3: So we're essentially at this point saying that $x(t)$ is the motion of object in time and $\dot x(t)$ is the velocity in time, but we don't make them dependent yet
Question 4: Then we vary. $\Delta S = \int_{t_1}^{t_2} \frac{\partial L}{\partial x} f(t) + \frac{\partial L}{\partial \dot x} \dot f(t)$. Here $f(t)$ and $\dot f(t)$ are those random numbers I was talking about and they can be any value at any moment. Even at this step, $f(t)$ and $\dot f(t)$ are still treated as randomly chosen and not dependent on each other.
Question 5: Somehow, now, we come and at this moment, we start treating $\dot f(t)$ to be derivative of $f(t)$. Check This - Why did we put the constraint only now that it now is derivative ? If we hadn't put this constraint, what would happen ?