Lagrangian and Hamiltonian Formalism on a Quantum Plane

TL;DR

This paper develops a formalism for Lagrangian and Hamiltonian mechanics on a quantum plane, introducing noncommutative geometry and differential calculus to generalize classical mechanics to quantum spaces.

Contribution

It constructs a framework for defining Lagrangian and Hamiltonian mechanics on a quantum plane with noncommuting coordinates and momenta, including deformed equations of motion and Poisson brackets.

Findings

Two differential calculi on the tangent quantum plane lead to different dynamics.

Deformed Hamilton's equations and Poisson brackets are derived for quantum plane mechanics.

Illustrations include a particle in a scalar potential and a first-order Lagrangian system.

Abstract

We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino, we construct two different differential calculi on $TQ_{q,p}$. These two differential calculi can in principle give rise to two different particle dynamics, starting from a single Lagrangian. For Hamiltonian mechanics, we define a phase space $T^*Q_{q,p}$ spanned by noncommuting particle coordinates and momenta. The commutation relations for the momenta can be determined only after knowing their functional dependence on coordinates and velocities. Thus these commutation relations, as well as the differential calculus on $T^*Q_{q,p}$, depend on the initial choice of Lagrangian. We obtain the deformed Hamilton's equations of motion and the deformed Poisson brackets, and their definitions also depend on our initial choice of Lagrangian. We illustrate these ideas for two sample Lagrangians. The first system we examine corresponds to that of a nonrelativistic particle in a scalar potential. The other Lagrangian we consider is first order in time derivatives