The κ-Deformed Calogero–Leyvraz Lagrangians and Applications to Integrable Dynamical Systems

The Calogero–Leyvraz Lagrangian framework, associated with the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proposed by Calogero and Leyvraz, has some special features. It is endowed with a Shannon “entropic” type kinetic energy term. In this paper, we carry out the constructions of the 2D Lotka–Volterra replicator equations and the N=2 Relativistic Toda lattice systems using this class of Lagrangians. We take advantage of the special structure of the kinetic term and deform the kinetic energy term of the Calogero–Leyvraz Lagrangians using the κ-deformed logarithm as proposed by Kaniadakis and Tsallis. This method yields the new construction of the κ-deformed Lotka–Volterra replicator and relativistic Toda lattice equations.