A Noether Thing

I've read up a little on Noether's theorem and whether it applies to scale-invariant Lagrangians. The answer seems to be either "yes" or "no".

The Feynman Lectures Vol. I, section 52, talks about symmetry and scaling. Feynman points out that Galileo discussed the physical problems of scaling in Two New Sciences. After reading a bit more about that, I think we are robbing students of a proper knowledge of Galileo's contributions by spending as much time as we do talking about his heresy trial. It makes it seem like he was little more than a prominent supporter of Copernicus, and it badly shortchanges him.

If you were a geeky child like me, at some point you probably ran into the question "If ants can carry fifty times their own weight, wouldn't it ROCK if we could make an ant the size of a car?" The soul-crushing answer is, no, it wouldn't - because a huge ant would be squished by its own weight. Specifically, its strength scales quadratically with body length, but its weight scales cubically, so it soon grows too heavy to walk, let alone to ferry little boys and girls around town, catching up miscreant adults in its pincers. This analysis (although explained in terms of Ariosto's poetic giants) was first proposed by G. It turns out that Galileo not only did not welcome them, but actually saved us from the looming prospect of Insect Overlords.

Be that as it may, Feynman explanation is Galileo's - scale-invariant physics doesn't work essentially because of the problems keeping ratios lined up (e.g., volume and surface area change at different rates, so air resistance is higher for smaller bodies.)

So that's the "no" answer. More on the "yes" answer when I understand it better....
- posted by Scott @ 5:24 PM