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

Round 1

Reviewer 1 Report

In this paper the author consider a special class of Lagrangians proposed by Calogero and Leyvraz. Using this new type of Lagrangians he derived the celebrated Lotka-Volterra and replicator equation. Then he generalized the construction of Calogero and Leyvraz and considered a dfferent type kinetic energy term based on cross entropy and constructed N = 2 relativistic Toda lattice system.

The main goal of this paper was to express all these celebrated equations in terms of logarithmic kinetic energy using the deformation of the entropic kinetic energy term. He used Kaniadakis-deformed logarithm and exponential functions to deform these Calogero-Leyvraz type Lagrangian to give a new formulation of κ-deformed Lotka-Volterra, replicator equation and N = 2 relativistic Toda lattice system. He also extended this deformation to Tsallis class and derived Tsallis-deformed equations. All the original equations can be recovered from the deformed systems when κ→0.

The paper is clearly written and wort of publication.

Author Response

Thank you for the comments.

Reviewer 2 Report

Calogero-Leyvraz Lagrangians, describing the motion of charged particle in a magnetic field with friction, are generalized using both Kaniadakis and Tsallis deformed logarithms. The Lagrange equations of motion are derived. This paper is, to my mind, suitable for publication in its present form.

There are several minor typos in the paper:

- page 5, section 2.1 "chargedparticle" should be replaced by "charged particle"

- After eq. 2.16, Linard --> Liénard

- Before eq. 2.17, the definition of y as dy/dt does not make sense

- Before eq. 2.26, something is missing after "Suppose we take \kappa"

- Corrollary 2.2., something is missing after "yields a one-parameter family of second-order".

- Before eq. 3.4., a parenthesis is missing : (3.4

- Before section 3.2, Hamiltonins --> Hamiltonians

- Before eq. 3.19, "in terms cosine" --> "in terms of cosine"

- Before eq. 3.22, Khler --> K\"ahler

- Equation (4.7) should be moved at the beginning of section 2.3 where Tsallis logarithm is used for the first time.

Author Response

See the attachment, please.

Author Response File: Author Response.pdf