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The Hamiltonian mechanics of exotic particles

Andrea Amoretti, Daniel K. Brattan, Luca Martinoia

TL;DR

This work addresses dynamics when local boost invariance is absent by formulating Hamiltonian mechanics on Aristotelian manifolds with absolute time and space. It develops invariant phase-space dynamics, a class of free Hamiltonians, and a Liouville theorem on constraint surfaces, then uses uplifted Aristotelian Killing vectors to generate conserved currents and a stress-energy–momentum complex. Through kinetic theory and hydrodynamics, the authors show that a boost-agnostic gas exhibits ideal hydrodynamics at leading order and that the ideal gas law $P=nT$ holds universally, independent of the dispersion relation $\tilde{H}(\vec{p}^{2})$. Collectively, the framework provides a geometric foundation for systems lacking boost symmetry, with applications to condensed matter, active matter, and optimization dynamics, and points toward future quantum extensions and generating functionals for conserved currents.

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.

The Hamiltonian mechanics of exotic particles

TL;DR

This work addresses dynamics when local boost invariance is absent by formulating Hamiltonian mechanics on Aristotelian manifolds with absolute time and space. It develops invariant phase-space dynamics, a class of free Hamiltonians, and a Liouville theorem on constraint surfaces, then uses uplifted Aristotelian Killing vectors to generate conserved currents and a stress-energy–momentum complex. Through kinetic theory and hydrodynamics, the authors show that a boost-agnostic gas exhibits ideal hydrodynamics at leading order and that the ideal gas law holds universally, independent of the dispersion relation . Collectively, the framework provides a geometric foundation for systems lacking boost symmetry, with applications to condensed matter, active matter, and optimization dynamics, and points toward future quantum extensions and generating functionals for conserved currents.

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.

Paper Structure

This paper contains 21 sections, 9 theorems, 301 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conservation laws according to Aristotle
  3. Summary of Aristotelian manifold
  4. Killing vectors on Aristotelian manifolds
  5. Charge, energy and momentum in Aristotelian geometries
  6. Application: collisions, kinetic theory and hydrodynamics
  7. Elastic scattering
  8. Kinetic theory
  9. Ideal hydrodynamics of particles with a single dispersion relation
  10. Developing Hamiltonian mechanics
  11. Phase space geometry
  12. Killing vectors on phase space
  13. "Free" Hamiltonians
  14. Embedding codimension one hypersurfaces of $M$ in $T^{*}M$
  15. Level sets of the Hamiltonian and Liouville's theorem
  16. ...and 6 more sections

Key Result

Theorem 1

There exists a canonical phase-space volume $\mathrm{vol}_{T^{*}M}$, defined in terms of the symplectic form $\Omega$, and given by which is conserved under the flow of any Hamiltonian vector field.

Figures (2)

  • Figure 1: An illustration of an elastic scattering between two hard sphere, distinguishable particles who initial momenta are at right angles to each other.
  • Figure 2: Energy density $\epsilon_u$, charge density $n$ and kinetic mass density $\rho$ as functions of the velocity $v$ in 2 and 3 dimensions for a Lifshitz and Galilean gas. The parameters used are $T=0.5$ and $\alpha=2m=2$.

Theorems & Definitions (25)

  • Definition 1: Aristotelian Killing vector
  • Theorem 1: Liouville's theorem on $T^{*}M$
  • proof
  • Definition 2: The tautological one-form/symplectic potential
  • Proposition 2: Properties of the uplift of a Killing vector field
  • proof
  • Definition 3: Poisson bracket
  • Lemma 1: Conservation of some free invariants
  • proof
  • Definition 4: Free Hamiltonian
  • ...and 15 more