# Lagrangian for a free particle

In Landau's Mechanics, if an inertial frame $\textit{K}$ is moving with an infinitesimal velocity $\textbf{ε}$ relative to another inertial frame $\textit{K'}$, then $\textbf{v}'=\textbf{v}+\textbf{ε}$. Since the equations of motion must have the 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 co-ordinates and time. Then he gave the formula $L'=L(v'^2)=L(v^2+2\textbf{v}\bullet\textbf{ε} + \textbf{ε}^2)$.
So my question is what does the sentence 'the equations of motion must have the same form in every frame' mean? Whether $L'(v'^2)=L(v'^2)$ or $L'(v'^2)=L(v^2)$? Why?
And what is the variable in the two Lagrangians,$\textbf{v}$ or $\textbf{v}'$?