A Landscape of Hamiltonian Phase Spaces: on the foundations and generalizations of one of the most powerful ideas of modern science
Abstract
In this thesis we revise the concept of phase space in modern physics and devise a way to explicitly incorporate physical dimension into geometric mechanics. A historical account of metrology and phase space is given to illustrate the disconnect between the theoretical physical models in use today and the formal treatment of units of measurement. Selfcontained presentations of local Lie algebras, Lie algebroids, Poisson manifolds, line bundles and Jacobi manifolds are given. A unitfree manifold is defined as a generic line bundle over a smooth manifold that we interpret as a manifold whose ring of functions no longer has a preferred choice of a unit. This point of view allows us to implement physical dimension into geometric mechanics. Unitfree manifolds are shown to share many of the core structure of the category of ordinary smooth manifolds: Cartesian products, derivations as tangent vectors, jets as cotangent vectors, submanifolds and quotients. This allows to reinterpret the notion of Jacobi manifold as the unitfree analogue of Poisson manifolds. With this new language we rediscover known results about Jacobi maps, coisotropic submanifolds, Jacobi products and Jacobi reduction. We give a categorical formulation of the loose term 'canonical Hamiltonian mechanics' by defining the notions of theory of phase spaces and Hamiltonian functor. Conventional configuration spaces are then replaced by line bundles, called unitfree configuration spaces, and, they are shown to fit into a theory of phase spaces with a Hamiltonian functor given by the jet functor. Motivated by the algebraic structure of physical quantities in dimensional analysis, we define dimensioned groups, rings, modules and algebras by implementing an addition operation that is partially defined. Jacobi manfolds are shown to have associated dimensioned Poisson algebras and dimensioned coisotropic calculus.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.08469
 Bibcode:
 2019arXiv191008469Z
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics
 EPrint:
 Thesis completed in the School of Mathematics of the University of Edinburgh under the supervision of Jose FigueroaO'Farrill