Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism

Abstract

We consider Feynman-Dyson's proof of Maxwell's equations using the Jacobi identities on the velocity phase space. In this paper we generalize the Feynman-Dyson's scheme by incorporating the non-commutativity between various spatial coordinates along with the velocity coordinates. This allows us to study a generalized class of Hamiltonian systems. We explore various dynamical flows associated to the Souriau form associated to this generalized Feynman-Dyson's scheme. Moreover, using the Souriau form we show that these new classes of generalized systems are volume preserving mechanical systems.

arXiv comment30 pages, few references have been added

References in corpus (36)

• On Foundation of the Generalized Nambu Mechanics
• A Gravity Theory on Noncommutative Spaces
• Non-Commutative Quantum Mechanics
• Quantum mechanics on the noncommutative plane and sphere
• Exotic galilean symmetry in the non-commutative plane, and the Hall effect
• Anyon wave equations and the noncommutative plane
• Newton's Second Law in a Noncommutative Space
• $\kappa$-Minkowski Spacetime Through Exotic "Oscillator"
• Thermodynamical properties of graphene in noncommutative phase-space
• Phase-space noncommutative formulation of Ozawa's uncertainty principle
• Conformal Invariance in noncommutative geometry and mutually interacting Snyder Particles
• Hydrogen atom in rotationally invariant noncommutative space
• Snyder dynamics in a Schwarzschild spacetime
• On the Landau system in noncommutative phase-space
• Classical dynamics on curved Snyder space
• Quantum mechanics with coordinate dependent noncommutativity
• Minimal coupling and Feynman's proof
• Magnetic Geodesic Flows on Coadjoint Orbits
• The Kepler problem in the Snyder space
• Non-commutative oscillator with Kepler-type dynamical symmetry
• Noncommutative configuration space. Classical and quantum mechanical aspects
• On The Classical Dynamics Of Charges In Noncommutative QED
• Feynman problem in the noncommutative case
• Bosonized supersymmetry of anyons and supersymmetric exotic particle on the non-commutative plane
• Noncommutativity in (2+1)-dimensions and the Lorentz group
• Wong's Equations and Charged Relativistic Particles in Non-Commutative Space
• Non-commutative mechanics and Exotic Galilean symmetry
• From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics
• Extended Space Duality in the Noncommutative Plane
• Closedness of orbits in a space with SU(2) Poisson structure
• Exotic Hill Problem: Hall motions and symmetries
• Noncommutative Phase Spaces by Coadjoint Orbits Method
• Generally covariant quantum mechanics on noncommutative configuration spaces
• Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
• Symmetries of field theories on the non-commutative plane
• The Euler-Lagrange Cohomology and General Volume-Preserving Systems