classical mechanics - Landau's argument for dependence of Lagrangian on magnitude of velocity

In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the particle.

This seemed fine. But he didn't give a reason as to why it must depend of $|v|^2$ and not on some $|v|^n$ where $n\neq2$.

Answer

The choice of the dependence is largely arbitrary at that point. In that chapter they just choose lagrangian as $L(v^2)$. If they chose it as $L^\prime(|v|^n)$ in equation $(3.1)$, they'd just have to say $L^\prime(|v|^n)=\frac12 m\sqrt[n]{|v|^n}^2$ in equation $(4.1)$.

Also, this choice is simple enough because it's merely a dot product of velocity with itself, which is the simplest scalar function of a vector.