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 aim to revise the fundamental concept of phase space in modern physics and to devise a way to explicitly incorporate physical dimension into geometric mechanics. To this end we begin with a nearly self-contained presentation of local Lie algebras, Lie algebroids, Poisson and symplectic manifolds, line bundles and Jacobi and contact manifolds, that elaborates on the existing literature on the subject by introducing the notion of unit-free manifold. We give a historical account of the evolution of metrology and the notion of phase space in order to illustrate the disconnect between the theoretical models in use today and the formal treatment of physical dimension and units of measurement. A unit-free manifold is, mathematically, simply a generic line bundle over a smooth manifold but our interpretation, that will drive several conjectures and that will allow us to argue in favor of its use for the problem of implementing physical dimension in geometric mechanics, is that of a manifold whose ring of functions no longer has a preferred choice of a unit. We prove a breadth of results for unit-free manifolds in analogy with ordinary manifolds: existence of Cartesian products, derivations as tangent vectors, jets as cotangent vectors, submanifolds and quotients by group actions. This allows to reinterpret the notion of Jacobi manifold found in the literature as the unit-free analogue of Poisson manifolds. With this new language we provide some new proofs for Jacobi maps, coisotropic submanifolds, Jacobi products and Jacobi reduction. We give a precise categorical formulation of the loose term 'canonical Hamiltonian mechanics' by defining the precise notions of theory of phase spaces and Hamiltonian functor, which in the case of conventional symplectic Hamiltonian mechanics corresponds to the cotangent functor. Conventional configuration spaces are then replaced by line bundles, what we call unit-free configuration spaces, and, after proving some specific results about the contact geometry of jet bundles, they are shown to fit into a theory of phase spaces with a Hamiltonian functor given by the jet functor. We thus find canonical contact Hamiltonian mechanics. Motivated by the algebraic structure of physical quantities in dimensional analysis, we redevelop the elementary notions of the theory of groups, rings, modules and algebras by implementing an addition operation that is only defined partially. In this formalism, the notions of dimensioned ring and dimensioned Poisson algebra appear naturally and we show that Jacobi manifolds provide a prime example. Furthermore, this correspondence allows to recover an explicit Leibniz rule for the Jacobi bracket, Jacobi maps, coisotropic submanifolds, and Jacobi reduction as their dimensioned Poisson analogues thus completing the picture that Jacobi manifolds are the direct unit-free analogue of ordinary Poisson manifolds.