  In this paper we will explore fundamental constraints on the evolution of
certain symplectic subvolumes possessed by any Hamiltonian phase space. This
research has direct application to optimal control and control of conservative
mechanical systems. We relate geometric invariants of symplectic topology to
computations that can easily be carried out with the state transition matrix of
the flow map. We will show how certain symplectic subvolumes have a minimal
obtainable volume; further if the subvolume dimension equals the phase space
dimension, this constraint reduces to Liouville's Theorem. Finally we present a
preferred basis that, for a given canonical transformation, has certain
minimality properties with regards to the local volume expansion of phase
space.
