The Hamiltonian mechanics of exotic particles

TL;DR

This paper develops a Hamiltonian framework on Aristotelian manifolds lacking boost symmetry, enabling analysis of systems in condensed matter and active matter with absolute time and space.

Contribution

It introduces a geometric Hamiltonian mechanics approach on Aristotelian manifolds, extending classical mechanics to systems without boost invariance.

Findings

Invariant phase space dynamics constructed

Generalized Liouville theorem established

Ideal hydrodynamics and gas law derived universally

Abstract

We develop Hamiltonian mechanics on Aristotelian manifolds, which lack local boost symmetry and admit absolute time and space structures. We construct invariant phase space dynamics, define free Hamiltonians, and establish a generalized Liouville theorem. Conserved quantities are identified via lifted Killing vectors. Extending to kinetic theory, we show that the charge current and stress tensor reproduce ideal hydrodynamics at leading order, with the ideal gas law emerging universally. Our framework provides a geometric and dynamical foundation for systems where boost invariance is absent, with applications including but not limited to: condensed matter, active matter and optimization dynamics.