- #1

gionole

- 281

- 24

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)

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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