Dynamics in a Noncommutative Space

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 manifold. The noncommutativity exists in the coordinates or the momentum planes embedded in the 4D cotangent manifold. This noncommu-tativity is reflected in the derivation of the first-order Lagrangians by exploiting the most general form of the Legendre transformation defined on the noncommutative (co-) tangent manifolds. It is very interesting to point out that 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 manifold 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.