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Intuitively this actually is very plausible, though you have to work your intuition a bit; fix a point in configuration space, this can be seen as a single point in something that locally looks like $\Bbb R^{2n}$ or $\Bbb R^{3n}$, but it is more convenient to think of $n$ distinct points of (something that locally looks like) $\Bbb R^2$ or $\Bbb R^3$. A ...

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