The κ -deformed entropic Lagrangians, Hamiltonian dynamics and their applications

This is a sequel to our previous paper (Guha in Phys J Plus 137:64, 2022) on Calogero–Leyvraz’s method to 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. In this paper we modify Calogero–Leyvraz’s entropic kinetic energy term using κ- deformed logarithm introduced by Tsallis and Kaniadakis to study generalized rate equation. We also derive new kind of fractional damped Liénard type equation which satisfies Chiellini integrability condition. Using deformed Legendre transformation we map the κ-deformed entropic Lagrangian to κ deformed Hamiltonian of Calogero–Leyvraz type. We study Hamiltonian mechanics using κ-deformed Hamiltonian, where the kinetic part is given by the κ- deformed exponential series. In particular, we give a Hamiltonian formulation for one parameter time-dependent balanced gain–loss system using κ-deformed logarithm, this reduces to ordinary balanced gain–loss system when κ→ 0. Finally, we demonstrate the connection between κ-deformed Calogero–Leyvraz Hamiltonian with the κ-deformed Lambert W_κ (or Lambert–Kaniadakis) W_κ function.