Dynamics of Symplectic SubVolumes

TL;DR

This paper investigates the evolution constraints of symplectic subvolumes in Hamiltonian phase space, linking symplectic topology invariants to practical computations, with implications for optimal control of mechanical systems.

Contribution

It introduces a connection between symplectic topology invariants and computational methods for analyzing phase space volume evolution, including minimal volume constraints and basis selection.

Findings

Certain symplectic subvolumes have minimal obtainable volume.

When subvolume dimension equals phase space, constraints reduce to Liouville's Theorem.

A preferred basis minimizes local volume expansion for canonical transformations.

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.