Measurand Spaces and Dimensioned Hamiltonian Mechanics

TL;DR

This paper generalizes Hamiltonian mechanics by replacing configuration spaces with line bundles, integrating physical units into the mathematical framework through Jacobi manifolds and contact structures.

Contribution

It introduces a novel framework that incorporates measurand spaces and units into Hamiltonian mechanics using line bundles and Jacobi manifold theory.

Findings

Successfully generalizes Hamiltonian mechanics with measurand spaces.

Provides technical results on contact structures in jet bundles.

Integrates units and physical dimensions systematically.

Abstract

In this paper we introduce a generalization of Hamiltonian mechanics that replaces configuration spaces, conventionally regarded simply as smooth manifolds, with line bundles over smooth manifolds. Classical observables are then identified with the sections of these (generically non-trivial) line bundles. This generalization, mathematically articulated with theory of Jacobi manifolds, is motivated by a conceptual revision of the mathematical foundations of the notion of measurand and unit of measurement in practical science. We prove several technical results for the contact structures present on jet bundles in order to argue that our proposal does indeed successfully generalize Hamiltonian mechanics while incorporating a systematic treatment of physical dimension and units.