Eur. Phys. J. Plus (2022) 137: 64
https://doi.org/10.1140/epjp/s13360-021-02285-z

Regular Article

Balanced gain-loss dynamics of particle in cyclotron with friction, -defomed logarithmic Lagrangians and fractional damped systems

Department of Mathematics, Khalifa University of Science and Technology, Zone-1, Main Campus, Abu Dhabi, United Arab Emirates

^a partha.guha@ku.ac.ae

Received: 2 October 2021
Accepted: 9 December 2021
Published online: 27 December 2021

Abstract

In this paper, we study the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proportional to the velocity, introduced by Calogero and Leyvraz (J Nonlinear Math Phys 6:147–154, 2019; 26:228–239, 2019). In general, the coefficients of the equations are time-dependent, we explore its connection with the Lorentz interaction and balanced loss and gain system. In particular, for time-independent case we map this motion in cyclotron with friction to PT-symmetric linear dimer model. We demonstrate that solutions of the equations follow from logarithmic Lagramgian often can be expressed in terms of Lambert W function. We modify Calogero–Leyvraz Lagrangians via -deformed logarithm as proposed by Tsallis and Kaniadakis, and show that the Euler–Lagrange equation yields fractional damped equations. Finally, we propose a Tsallis -defomed logarithmic Lagrangian and show this yields one parameter family of dissipative equations and also their fractional damping counterpart. We complete our paper with a modest outlook showing that the Calogero–Leyvraz is a velocity dependent curl force analog of nonconservative position dependent curl force as proposed by Berry and Shukla (J Phys A 45:305201, 2012; Proc R Soc A 471:20150, 2015).