# n-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance

## TL;DR

This paper demonstrates that for multidimensional position-dependent mass systems, Newtonian invariance is more reliable than Euler-Lagrange invariance, enabling exact solutions for various nonlinear PDM oscillators through a nonlocal transformation.

## Contribution

It introduces the necessity of Newtonian invariance over Euler-Lagrange invariance in PDM systems and provides exact solutions for multiple types of nonlinear PDM oscillators.

## Key findings

- Newtonian invariance is more comprehensive than Euler-Lagrange invariance in PDM systems.
- Exact solutions for various nonlinear PDM oscillators were obtained.
- A nonlocal space-time transformation was used to derive solutions.

## Abstract

We argue that, under multidimensional position-dependent mass (PDM) settings, the Euler-Lagrange textbook invariance falls short and turned out to be vividly incomplete and/or insecure for a set of PDM-Lagrangians. We show that the transition from Euler-Lagrange component presentation to Newtonian vector presentation is necessary and vital to guarantee invariance. The totality of the Newtonian vector equations of motion is shown to be more comprehensive and instructive than the Euler-Lagrange component equations of motion (they do not run into conflict with each other though). We have successfully used the Newtonian invariance amendment, along with some nonlocal space-time point transformation recipe, to extract exact solutions for a set of n-dimensional nonlinear PDM-oscillators. They are, Mathews-Lakshmanan type-I PDM-oscillators, power-law type-I PDM-oscillators, the Mathews-Lakshmanan type-II PDM-oscillators, the power-law type-II PDM-oscillators, and some nonlinear shifted Mathews-Lakshmanan type-I PDM-oscillators.