Why Lagrangian is $L=\frac{1}{2}mv^2$ and not $mv^2$ for a free particle in an intertial frame? Both are proportional to the square of velocity

Question

Landau writes the Lagrangian of a free particle in a second inertial frame as $$L(v'^{2})=L(v^2)+\frac{\partial L}{\partial v^2}2\textbf{v}\cdot{\epsilon},$$ and then it's written that the Lagrangian is in this case proportional to the square of velocity , and we write it as: $$L=\frac{1}{2}mv^2,$$ my question is: why not just $L=mv^2$? The latter case is proportional to the square of velocity as well.

Answer 1

Both would work. It is just a matter of convention. Notice that if $x_0$ is an extremum of the function $S(x)$, it is also an extremum of $\alpha \cdot S(x)$ for constant $\alpha \neq 0$.