I have a thin eclectic volume, "Mechanics," by Landau and Lifshitz
(Course of Theoretical Physics, v1).
In the discussion of the Lagrangian and equations of motion, the
concept of the inertial frame is introduced; and then:
"... a frame of reference can always be chosen in which space is
homogeneous and isotropic and time is homogeneous. This is called an
inertial frame... We can now draw some immediate inferences
concerning the form of L of a free particle in an inertial frame. The
homogeneity of space and time implies that L cannot contain
explicitly either the radius vector r of the particle or time t,
i.e., L must be a function of velocity v only. Since space is
isotropic, L must also be independent of the direction of v, and is
therefore a function only of its magnitude, i.e. v^2."
Ergo: L = L(v^2).
This is the closest I've seen to an argument from first principles
for the form of L of a free particle, that is, why KE is proportional
to v^2. I like the idea pedagogically of an argument that doesn't
resort to blatant/circular assertions about the form of KE. However,
Landau/Lifshitz do not expound further either, beyond these few
words. I think I understand the second part, essentially that the
isotropy of space cannot support a preferred direction, but it's
unclear to me how the homogeneity of space and time lead to L being a
function of v alone. I'm guessing that spatial homogeneity implies
that whatever happens in "this cubic meter" must happen identically
in "the next cubic meter," but the a priori necessity of temporal
homogeneity is lost on me - obviously, conserved quantities are
dependent on constancy over time, but his argument seems to precede
his discussion of constants of motion.