Hamiltonian and Lagrangian Dynamics in a Noncommutative Space

TL;DR

This paper explores the Hamiltonian and Lagrangian formulations of a 2D physical system within a 4D (non-)commutative phase space, highlighting how noncommutativity influences the dynamics and formalism structure.

Contribution

It introduces a consistent approach to formulating Hamiltonian and Lagrangian dynamics in noncommutative phase space using symplectic structures, revealing invariance in second-order Lagrangians.

Findings

Noncommutativity can be in coordinate or momentum planes.

Second-order Lagrangian remains unchanged despite noncommutativity.

Connection established between noncommutative dynamics and quantum group structures.

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.