Position-dependent mass Lagrangians: nonlocal transformations, Euler-Lagrange invariance and exact solvability

TL;DR

This paper introduces a non-local transformation approach for position-dependent mass Lagrangians, enabling their mapping to constant-mass systems, and explores their invariance, linearization, and exact solutions through various oscillator examples.

Contribution

It presents a novel non-local point transformation framework for PDM Lagrangians, establishing invariance conditions and demonstrating exact solvability for complex oscillators.

Findings

Mapped PDM oscillators to constant-mass systems

Reproduced Mathews-Lakshmanan nonlinear oscillators

Provided examples of exact solutions and invariance conditions

Abstract

A general non-local point transformation for position-dependent mass Lagrangians and their mapping into a "constant unit-mass" Lagrangians in the generalized coordinates is introduced. The conditions on the invariance of the related Euler-Lagrange equations are reported. The harmonic oscillator linearization of the PDM Euler-Lagrange equations is discussed through some illustrative examples including harmonic oscillators, shifted harmonic oscillators, a quadratic nonlinear oscillator, and a Morse-type oscillator. The Mathews-Lakshmanan nonlinear oscillators are reproduced and some "shifted" Mathews-Lakshmanan nonlinear oscillators are reported. The mapping of an isotonic nonlinear oscillator into a PDM deformed isotonic oscillator is also discussed.