A Landscape of Hamiltonian Phase Spaces: on the foundations and generalizations of one of the most powerful ideas of modern science

TL;DR

This thesis revises the concept of phase space in physics by integrating physical dimensions into geometric mechanics through the novel framework of unit-free manifolds, offering a categorical and algebraic foundation for Hamiltonian systems.

Contribution

It introduces the concept of unit-free manifolds, generalizes Jacobi manifolds, and develops a categorical formulation of Hamiltonian mechanics incorporating physical dimensions.

Findings

Unit-free manifolds share core structures with smooth manifolds.

Jacobi manifolds are interpreted as unit-free analogues of Poisson manifolds.

Dimensioned algebraic structures model physical quantities with partial addition.

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. Self-contained presentations of local Lie algebras, Lie algebroids, Poisson manifolds, line bundles and Jacobi manifolds are given. A unit-free 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. Unit-free 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 unit-free 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 unit-free 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.