I'm reading volume n. 1 of "Course of Theoretical Physics" by Landau,
Scope: Lagrange function of a material point in free motion, in an inertial system.
What: two inertial systems, K ad K', the second moving with velocity $ v' = v + \epsilon $. We have:
$ L' = L(v'^2) = L (v^2 + 2v\epsilon+\epsilon^2) = L(v^2)+\frac{\partial L}{\partial v^2} (2v\epsilon + \epsilon^2) + O(v\epsilon) = L(v^2)+\frac{\partial L}{\partial v^2}2v\epsilon + O(v\epsilon) $.
First: In the book the error estimation is not specified. Is my addition correct?
Second: The book proceedes by saying that $ \frac{\partial L}{\partial v^2}2v\epsilon $ is a total derivative with respect to time only if there's a linear dependence to the velocity $v$, therefore $\frac{\partial L}{\partial v^2}$ is independent to the velocity $v$. And this is equivalent to saying that the Lagrange function is proprortional to $v$, as following $L = av^2$, where $a$ is a constant. Why?
Can't get it. Any help?