Title: Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and Quantum Mechanics,
Abstract: In the usual approaches to mechanics (classical or quantum) the primary
object of interest is the Hamiltonian, from which one tries to deduce the
solutions of the equations of motion (Hamilton or Schrödinger). In the
present work we reverse this paradigm and view the motions themselves as being
the primary objects. This is made possible by studying arbitrary phase space
motions, not of points, but of (small) ellipsoids with the requirement that the
symplectic capacity of these ellipsoids is preserved. This allows us to guide
and control these motions as we like. In the classical case these ellipsoids
correspond to a symplectic coarse graining of phase space, and in the quantum
case they correspond to the "quantum blobs" we defined in previous work, and
which can be viewed as minimum uncertainty phase space cells which are in a
one-to-one correspondence with Gaussian pure states.