Question on Poincare Recurrence Theorem

[kakarukeys]

Poincare Recurrence Theorem states that:
"If a flow preserves volume and has only bounded orbits then for each open set there exist orbits that intersect the set infinitely often."

But it does not imply (does it?) that
"In hamiltonian system with bounded phase space, all trajectories will eventually return arbitrarily close to the original starting point."

Only some not all trajectories will do so. When we consider a small neighbourhood of the starting point, and by the theorem, there exist some orbits (not all) that intersect the set later.

 

[dextercioby]

Yes,Poincaré's result was the main counterargument physicists found to the veridicity of Boltzmann's theory of kinetics...

Daniel.

 

[charng]

Your statement is correct. The Poincare Recurrence Theorem does not guarantee that all trajectories in a Hamiltonian system with a bounded phase space will return arbitrarily close to the original starting point. It only guarantees that there will exist some orbits that intersect a given open set infinitely often. This means that there will be some trajectories that do return close to the starting point, but not necessarily all of them. The theorem also does not specify when or how often these returns will occur. Additionally, the theorem assumes that the flow preserves volume, which may not always be the case in a Hamiltonian system. Therefore, while the Poincare Recurrence Theorem is a useful tool for understanding the behavior of dynamical systems, it is important to consider its limitations and the specific conditions under which it applies.