  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.
