Independence of position and velocity in Lagrangian from the point of view of physics? [duplicate]

Question

I would like to continue discussion from my previous post on time dependence of lagrangian Time dependence of the Lagrangian of a free particle?. I have also read this old post Why does calculus of variations work?

I think I understand how Lagrangian treats position and velocity as independent variables from the point of view of calculus of variations. I think velocity only really deals with "change in position" and not the position itself, and so my intuition is like this should be one of the reasons we have to treat position and velocity as independent from the physics point of view as well, and not just so because calculus of variations treats Lagrangian as a function of some variables (x,y,z). Even the change in velocity is just related to change in position in the relation below and not the position itself. $$\delta \dot q = {d \over dt} \delta q$$

So I think that not only position and velocity initial conditions are independent, but position and velocity are independent along any trajectory too. Is this why Lagrangian is specifically defined with position and velocity independent ? Also, position forms an affine space and so it doesn't make sense to me how can anything be dependent on it unless we define a frame (or origin). And we treat Lagrangian as frame independent too. Does this make any sense ? I just want to make things precise here. I hope my mind is not messing with me here.

Thank you

Answer 1

You say

but position and velocity are independent along any trajectory too.

No, a trajectory is defined as a given vector-function $\vec r(t)$ from which follows $\vec v(t) = d\vec r(t)/dt$. So, given a trajectory, the vector-function $\vec v(t)$ depends clearly on $\vec r(t)$ as being its derivative.

Now, forget the trajectories. You have an integral

$$\int_{\vec r_1}^{\vec r_2} L(\vec r, \vec v, t) dt.$$

Did someone until this point said that the integral is taken about a precise trajectory, a known trajectory? No!

1) Then pick a time $t$.

2) Do you know at this time the vector-function $\vec r$? No! Then pick at random a $\vec r$.

3) Do you know for this $\vec r$, the $\vec v$? No! At the position $\vec r$, the vector $\vec v$ may point in whatever direction. Then pick a vector $\vec v$.

By these three selections you chose a segment of trajectory which passes at the time $t$ through the position $\vec r$ ang continues for an interval $dt$ in the direction $\vec v$.

So, you have a 7-fold infinity of choices of trajectories. So, to summarize, only if you knew a well defined trajectory, $\vec v$ were dependent on $\vec r$, and both on $t$.

From now on, I recommend to look in your reference at the answer of grizzly adam. You can see that the concept of trajectory is not even introduced in the beginning of the proof of the E-L equation, only later.

About frame, yes, is it tacitly understood, at least in classical mechanics, that we take a specific frame. Anyway, it is better to understand the things classically, and only then to pass to relativity if you need.