Isochronous $n$-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability

TL;DR

This paper investigates the properties of $n$-dimensional nonlinear PDM oscillators, focusing on their isochronicity, invariance, and exact solvability, revealing limitations in linearizability for higher dimensions and proposing alternative invariances.

Contribution

It demonstrates that linearizability is only achievable for one-dimensional PDM oscillators and introduces alternative invariances for higher dimensions, providing exact solutions for specific isochronous systems.

Findings

Linearizability only possible for $n=1$

Euler-Lagrange invariance is incomplete for $n extgreater 1$

Alternative invariances enable exact solutions for certain systems

Abstract

Within the standard Lagrangian settings (i.e., the difference between kinetic and potential energies), we discuss and report isochronicity, linearizability and exact solubility of some $n$-dimensional nonlinear position-dependent mass (PDM) oscillators. In the process, negative the gradient of the PDM-potential force field is shown to be no longer related to the time derivative of the canonical momentum, $\mathbf{p}% =m\left( r\right) \mathbf{\dot{r}}$, but it is rather related to the time derivative of the pseudo-momentum, $\mathbf{\pi }\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$ (i.e., Noether momentum). Moreover, using some point transformation recipe, we show that the linearizability of the $n$-dimensional nonlinear PDM-oscillators is only possible for $n=1$ but not for $n\geq 2$. The Euler-Lagrange invariance falls short/incomplete for $n\geq 2$ under PDM settings. Alternative invariances are sought, therefore. Such invariances, like \emph{Newtonian invariance} of Mustafa \cite{42}, effectively authorize the use of the exact solutions of one system to find the solutions of the other. A sample of isochronous $n$-dimensional nonlinear PDM-oscillators examples are reported.