Hamiltonian and Lagrangian Dynamics in a Noncommutative Space

Abstract

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists {\it equivalently} in the coordinate or the momentum planes embedded in the 4D cotangent manifolds. The signature of this noncommutativity is reflected in the derivation of the first-order Lagrangians where we exploit the most general form of the Legendre transformation defined on the (non-)commutative (co-)tangent manifolds. The second-order Lagrangian, defined on the 4D {\it tangent manifold}, turns out to be the {\it same} irrespective of the noncommutativity present in the 4D cotangent manifolds for the discussion of the Hamiltonian formulation. A connection with the noncommutativity of the dynamics, associated with the quantum groups on the q-deformed 4D cotangent manifolds, is also pointed out.

LaTeX, 12 pages, minor changes in the title and text, references expanded, version to appear in Mod. Phys. Lett. A

References in corpus (3)

• Testing spatial noncommutativiy via the Aharonov-Bohm effect
• Noncommutative Synchrotron
• Deformed Traces and Covariant Quantum Algebras for Quantum Groups $G{L}_{qp}(2)$ and $G{L}_{qp}(1|1)$

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• Landau levels in a 2D noncommutative space: matrix and quaternionic vector coherent states
• Interacting Relativistic Particle: Time-Space Noncommutativity And Symmetries
• Duality between the coordinates and wave functions on noncommutative space
• Cohomological Operators and Covariant Quantum Superalgebras