  We present a systematic treatment of line bundle geometry and Jacobi
manifolds with an application to geometric mechanics that has not been noted in
the literature. We precisely identify categories that generalise the ordinary
categories of smooth manifolds and vector bundles to account for a lack of
choice of a preferred unit, which in standard differential geometry is always
given by the global constant function $1$. This is what we call the `unit-free'
approach. After giving a characterisation of local Lie brackets via their
symbol maps we apply our novel categorical language to review Jacobi manifolds
and related notions such as Lichnerowicz brackets and Jacobi algebroids. The
main advantage of our approach is that Jacobi geometry is recovered as the
direct unit-free generalisation of Poisson geometry, with all the familiar
notions translating in a straightforward manner. We then apply this formalism
to the question of whether there is a unit-free generalisation of Hamiltonian
mechanics. We identify the basic categorical structure of ordinary Hamiltonian
mechanics to argue that it is indeed possible to find a unit-free analogue.
This work serves as a prelude to the investigation of dimensioned structures,
an attempt at a general mathematical framework for the formal treatment of
physical quantities and dimensional analysis.
