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Hamiltonian and Lagrangian Dynamics in a Noncommutative Space

R. P. Malik
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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.

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R. P. Malik. 2003-10-22. Hamiltonian and Lagrangian Dynamics in a Noncommutative Space. https://doi.org/10.1142/s0217732303012350

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