Landau's inertial frame logic

[gionole]

I had an interesting thought.

Let's only look at the free particle scenario.

We derive euler lagrange even without the need to know what exactly $L$ is (whether its a function of kinetic energy or not) - deriving EL still can be done. Though, because in the end, we end up with such EL($\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q} = 0$), we see that $L$ couldn't have been a function of $\dot q$ which depends on $t$, because if $\dot q$ depends on $t$, euler lagrange couldn't be applied to it as EL derivates $L$ wrt to $\dot q$.

So at this time, we know $L$ is a function of $v$ in which $v$ doesn't depend on $t$.

Then Landau tries to come up with what $L$ is. in the $K'$ inertial frame, he shows that $L' = L(v^2) + \frac{dL}{dv^2}2v\epsilon$. Everything is clear till now, but then he changes $v$ into $\frac{dr}{dt}$. How can he do that if the initial assumption is that $v$ and $q$ are not a function of $t$ in $L$ ? (I know that adding total time derivative doesn't change EOM, but this question is not about this)

 

[gionole]

I think I figured out the logic in my head.

By that, he doesn't say that $L'$ is a function of $v, q$ which depend on $t$ - he doesn't say this. He just shows that adding total time derivative doesn't change EOM.