Dual Spaces, Lie Groups and Conservation Laws

This is really impressive:
 
 

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I wonder what YouTube will suggest in the auto-reply to my comment:

@VisualMath  1:20 on compactness and averaging, ... and on smooth geometric structure in euclidean spaces. I have this intuition about variational calculus that I am also hoping to be able to explain once I have a nice representation of differentiable manifold, or maybe it'll work the other way around and I'll be able to get a nice representation if I follow this intuition? Maybe there's a way to find it by converging from both directions under some duality assumptions? The intuition is that in some sense the physical reason why the principle of least action works (and why Lagrange called it least action, not just stationary action) is that if we are observing some system and trying to deduce laws of motion for it, then we want to only select those paths through the configuration space which are the ones the system follows spontaneously, i.e. by 'natural flows' of energy, rather than un-natural ones where the experimenter bumped the measuring device or something. So we want to select the ones where the 'action' of the system is the least in some sort of causal sense. And this amounts to assuming the conservation laws of energy/momentum etc. because we are rejecting spurious inputs/losses of energy/momentum etc. And it's well-understood that these conservation laws are closely related to symmetries from Nöther's theorem. But in real physical systems we don't have perfect information from measurements so we want to be able to design experiments which are well behaved observationally under these variations in measurement accuracy, in that the models converge in stable ways to some structure which fits the observation best. This is a better way to proceed, I think, than just assuming things like point masses moving through real euclidean vector spaces under central forces and calling that physical determinism! So what I am saying that a proper physical model would be one where the averaging is something we have to do because of limited measurement precision, and because of limited precision in calculating numerical solutions, and that the smooth manifold emerges from this, stochastically, rather than being some Platonic theatre in which physics plays out to us as observers whether we know about it or not. It's more like the show only happens if we turn up to watch. 

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