Dynamics of Symplectic SubVolumes

Abstract

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.