Why does Isotropy of L imply L(v^2)?
I'm reading the first edition of Mechanics by Landau et al, published in 1960. Just before equation 3.1 on page 5 it says exactly this:
"Since space is isotropic, the Lagrangian must also be independent of the direction of v, and is therefore a function only of it's magnitude, i.e. of v(bold)^2 = v(italic)^2:
L = L(v(italic)^2) (3.1)"
This seems very cryptic to me since the magnitude is sqrt(v(bold)^2) =
Could someone fill in the missing details for me please?