I Landau's inertial frame logic

AI Thread Summary
The discussion focuses on the derivation of the Euler-Lagrange equations in the context of free particles, emphasizing that the Lagrangian, denoted as L, does not depend on time. It is established that L must be a function of velocity v, which is independent of time, leading to the conclusion that the Euler-Lagrange equations can still be applied. Landau's approach introduces the transformation of L in the K' inertial frame, where he modifies L by substituting v with dr/dt. This raises a question about the validity of this substitution given the initial assumption that v and q are not time-dependent. Ultimately, the discussion clarifies that Landau's addition of a total time derivative does not alter the equations of motion, maintaining the integrity of the analysis.
gionole
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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)
 
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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.
 
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