Why L=L（v^2) in inertial reference system?

[Dr turtle]

TL;DR Summary

Why Landau pointed out that Lagrange function shall only be affected by v square in inertial reference system?

Why he said that beacause space's propertiy is the same in both direction, so L=L(v^2), or do I misunderstand him incorrectly?
btw this conclusion appears in somewhere like page 5 and its about Galilean principle of relativity.

 

[anuttarasammyak]

The direction should not matter, so only magnitude of velocity |v| should matter.
$$f(|v|)=f(\sqrt{v^2})=g(v^2)$$
So we can say only v^2 matters.

 

[Dr turtle]

That's really helpful, lots of thanks