Non-commutative mechanics and Exotic Galilean symmetry

TL;DR

This paper employs the Lagrange-Souriau 2-form formalism to derive Hamiltonian systems with non-commutative coordinates, revealing a broad class of models with applications across various phenomenological contexts.

Contribution

It introduces a systematic approach using Lagrange-Souriau 2-forms to generate Hamiltonian systems with non-commutative geometry, expanding the theoretical framework.

Findings

Non-commutative position coordinates naturally arise in the formalism.

The approach covers a wide range of phenomenological models.

Explicit examples demonstrate the formalism's applicability.

Abstract

In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological contexts are covered. The non-commutativity of the particle position coordinates are a natural consequence. Some explicit examples are considered.