Lagrangian formulation of some q-deformed systems

TL;DR

This paper develops a Lagrangian framework for q-deformed physical systems, deriving q-deformed equations of motion and symmetries, including gauge invariance and reparametrization, for free particles and oscillators.

Contribution

It introduces a q-deformed Legendre transformation and constructs Lagrangians for q-deformed systems, highlighting their gauge and reparametrization properties.

Findings

Derived q-deformed Legendre transformation for a quantum line

Constructed Lagrangians for q-deformed harmonic oscillator and relativistic particle

Identified conditions for gauge symmetry and reparametrization invariance in q-deformed systems

Abstract

In the framework of Lagrangian formulation, some q-deformed physical systems are considered. The q-deformed Legendre transformation is obtained for the free motion of a non-relativistic particle on a quantum line. This is subsequently exploited to obtain the Lagrangians for the q-deformed harmonic oscillator and q-deformed relativistic free particle. The Euler-Lagrange equations of motion are derived in a consistent manner with the corresponding Hamilton's equations. The Lagrangian for the q-deformed relativistic particle is endowed with the q-deformed gauge symmetry and reparametrization invariance which are shown to be equivalent only for $ q = \pm 1$.