#Physicsfactlet (52) (part 1)
In a Hamiltonian system, if you take an ensemble of starting conditions and follow their evolution, at any time the points will occupy the same volume (in phase space), although the shape of this volume can change (Liouville's theorem).pic.twitter.com/KVJqB9prKN
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I am afraid that a clear explanation of this part would be beyond my capabilities

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Take a bounded smooth oriented surface in higher dimensional phase space. It has a well defined _oriented surface area_: the integral of the symplectic form over that surface. Evolve the surface with Hamiltonian mechanics. Then its _oriented surface area_ will remain constant!
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