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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.