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I'm trying to follow an argument in Landau's Mechanics. The argument concerns finding the Lagrangian of a free particle moving with velocity v relative to an inertial frame K. (of course L=1/2 mv^2, which is what we have to find). I'll state the points of the argument:

(0) It has already been argued that the Lagrangian relative to an intertial frame K must be of the form L(v^2) (space is homogeneous and iostropic).

(1) If an inertial frame K is moving with infinitesimal velocity e relative to another inertial frame K', the Lagrangian L' must be of the same form because the equations of motion are unchanged under Galilean transformations.

**(2) So the Lagrangian L' wrt K' must differ by L by at most a time derivative of some f(q,t).**

(3) L' = L(v'^2) = L(v^2 + 2v*e + e^2) which is to first-order [tex]L(v'^2) = L(v^2) + (\partial L/\partial v^2) 2 v\cdot e[/tex]

**(4) The second term in the last equation is a total time derivative only if it is a linear function of the velocity v. Hence [tex]\partial L/\partial v^2[/tex] is independent of the velocity. I.e. the Lagrangian is proportional to the square of the velocity.**

I'm having trouble with (2) and (4).

Specifically, my question for (2) is that it has been proven L and L' differing by a time derivative of some f(q,t) (q is a vector of generalized coordinates) does not change the solutions of the equations of motion, but the other way around. Thus 'must differ' in (2) isn't true. I guess 'allowed to differ' is more correct.

My question for (4) is that I don't get it. :)

Thanks

-evoluciona