Two-dimensional position-dependent mass Lagrangians; Superintegrability and exact solvability

TL;DR

This paper extends the concept of position-dependent mass Lagrangians to two dimensions, demonstrating mappings to exactly solvable systems and exploring superintegrability and sub-superintegrability in PDM-oscillators.

Contribution

It introduces a two-dimensional framework for PDM-Lagrangians and their mappings to solvable systems, highlighting superintegrability properties and specific oscillator examples.

Findings

Mapped superintegrable linear oscillators to sub-superintegrable PDM-oscillators.

Identified conditions for invariance of Euler-Lagrange equations.

Provided explicit examples of PDM-oscillators with superintegrability.

Abstract

The two-dimensional extension of the one-dimensional PDM-Lagrangians and their nonlocal point transformation mappings into constant unit-mass exactly solvable Lagrangians is introduced. The conditions on the related two-dimensional Euler-Lagrange equations' invariance are reported. The mappings from superintegrable linear oscillators into sub-superintegrable nonlinear PDM-oscillators are exemplified by, (i) a sub-superintegrable Mathews-Lakshmanan type-I PDM-oscillator, for a PDM-particle moving in a harmonic oscillator potential, (ii) a sub-superintegrable Mathews-Lakshmanan type-II PDM-oscillator, for a PDM-particle moving in a constant potential, and (iii) a sub-superintegrable shifted Mathews-Lakshmanan type-III PDM-oscillator, for a PDM-particle moving in a shifted harmonic oscillator potential. Moreover, the superintegrable shifted linear oscillators and the isotonic oscillators are mapped into a sub-superintegrable PDM-nonlinear and a sub-superintegrable PDM-isotonic oscillators, respectively.