Lecture 16 - Phase Volume Preservation and Recurrence

Now we will leave circle and toral maps for a bit and change our focus a bit.  We will stay on this notion of recurrence, however, but in a different fashion.  Instead of simply looking for a particular orbit to exhibit certain behavior, let's look at families of orbits simultaneously.  In particular, for a flow or a map, let's look at the orbit of a small open set in state (phase space), as in consider all points in a small open set to be the initial points and follow all of their orbits at once. 

Today we will introduce the notion of incompressability of a flow, which is also called phase volume preservation.  All of our toral translations and flows to date preserved volume, as do all isonetries, but contractions do not.  For a flow, this is related to divergence of the vector field of the flow, and we show that Newton;s Second Law of Motion provides a rich set of examples, along with exact differential equations. 

Then we show that the ramifications of volume preservation in a closed finite volume state space has interesting implications leading to the Poincare Recurrence Theorem.  We will end with a discussion of why this theorem holds. 

We will pass through some sections of passing interest that we will not explore, like Lagrange and Hamilton formulations of equations of motion and such.  They are related and of good general interest.  But they will not be a focus in this set of lectures. 

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