2003/02/28 by R. P. Malik · 12 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Classical mechanics #Covariant Hamiltonian field theory #Dynamics (music) #Hamiltonian (control theory) #Hamiltonian mechanics #Hamiltonian system #Lagrangian #Mathematical physics #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Phase space #Physics #Quantum mechanics #Space (punctuation) #Theoretical physics #hep-th
paper · pdf · doi:10.1142/s0217732303012350
published in Modern Physics Letters A 18(39), 2795-2806 (World Scientific) · LaTeX, 12 pages, minor changes in the title and text, references expanded, version to appear in Mod. Phys. Lett. A
arxiv created 2003/10/22 · openalex publication_date 2003/12/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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 tangent manifold, turns out to be the 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.