I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.

 1. **Proving that a free particle moves with a constant velocity in an inertial frame of reference** ($\S$3. Galileo's relativity principle). The proof begins with explaining that the Lagrangian must only depend on the speed of the particle ($v^2={\bf v}^2$):
$$L=L(v^2).$$
Hence the Lagrance's equations will be
$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\bf v}}\right)=0,$$
so
$$\frac{\partial L}{\partial {\bf v}}=\text{constant}.$$
And this is where the authors say

  > Since $\partial L/\partial \bf v$ is a function of the velocity only, it follows that
$${\bf v}=\text{constant}.$$

  **Why so?** I can put $L=\|{\bf v}\|=\sqrt{v^2_x+v^2_y+v^2_z}$. Then
$$\frac{\partial L}{\partial {\bf v}}=\frac{2}{\sqrt{v^2_x+v^2_y+v^2_z}}\begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix},$$
which will remain a constant vector $\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$ as the particle moves with an arbitrary non-constant positive $v_x$ and $v_y=v_z=0$. Where am I wrong here? If I am, how does one prove the quoted statement?

 2. **Proving that** $L=\frac{m v^2}2$ ($\S$4. The Lagrangian for a free particle). The authors consider an inertial frame of reference $K$ moving with a velocity ${\bf\epsilon}$ relative to another frame of reference $K'$, so ${\bf v'=v+\epsilon}$. Here is what troubles me:
> Since the equations of motion must have **same form** in every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, **only by the total time derivative of a function of coordinates and time** (see the end of $\S$2).

  First of all, what does **same form** mean? I think the equations should be the same, but if I'm right, why wouldn't the authors write so?
  Second, it was shown in $\S$2 that adding a total derivative will not change the equations. There was nothing about total derivatives of time and coordinates being **the only functions**, adding which does not change the equations (or their *form*, whatever it means). Where am I wrong now? If I'm not, how does one prove the quoted statement and why haven't the authors done it?

---
P. S. Could you recommend any textbooks on analytical mechanics? I'm not very excited with this one. Seems to hard for me.