Poincare recurrence

The trajectory will return to any region A, (A)>0.

Proof:


Assume that the trajectory does not return, then there will be no points that are members of both f-n(A) and A for any value of n>0

In particular for n > m

as a point in this intersection would return to this intersection after n-m steps.

If is the volume of the smallest of the sets f-NA,...,f-1A.



yields contradiction with finite region in phase-space.

Poincare recurrence is for the phase-point, not for the distribution of phase-points.

The proof nicely illustrates the situation typical in much of analytical mechanics. It shows that any Hamiltoian system has a some property, in this case recurrence, while the proof gives no hint of closely related question, e.g the time it will take before the recurrence will happen.

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Author Per Stoltze stoltze@fysik.dtu.dk