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

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 κ-deformed logarithm as proposed by Tsallis and Kaniadakis, and show that the Euler–Lagrange equation yields fractional damped equations. Finally, we propose a Tsallis κ-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).