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.

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.

Equations

$$L_{{}_I}\left(\vec{r},\vec{\upsilon };t\right)=\frac{1}{2}{m}_{\circ }\sum _{j=1}^n{m}_{{}_j}\left(x_{{}_j}\right){\dot{x}}_{{}_j}^2-V_{{}_I}\left(\vec{r}\right);j=1,2,\cdots ,n\in \mathbb{N},$$

$$L_{{}_{II}}\left(\vec{r},\vec{\upsilon };t\right)=\frac{1}{2}{m}_{\circ }m\left(\vec{r}\right)\sum _{j=1}^n{\dot{x}}_{{}_j}^2-V_{{}_{II}}\left(\vec{r}\right),$$

$$\vec{r}=\sum _{j=1}^n{x}_{{}_j}{\hat{x}}_{{}_j},\vec{\upsilon }=\frac{d\vec{r}}{dt}=\sum _{j=1}^n{\dot{x}}_{{}_j}{\hat{x}}_{{}_j},{\dot{x}}_{{}_j}=\frac{d{x}_{{}_j}}{dt}.$$

$$L\left(\vec{q},\overrightarrow{\tilde{q}};\tau \right)=\frac{1}{2}{m}_{\circ }\sum _{j=1}^n{\tilde{q}}_{{}_j}^2-V(\vec{q});{\tilde{q}}_{{}_j}=\frac{d{q}_{{}_j}}{d\tau };j=1,2,\cdots ,n,$$

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{x}}_i}\right)-\frac{\partial L}{\partial x_i}=0;i=1,2,\cdots ,n\in \mathbb{N},$$

$${\ddot{x}}_{{}_i}+\left(\frac{{\dot{m}}_{{}_i}\left(x_i\right)}{2m_{{}_i}\left(x_i\right)}\right){\dot{x}}_i+\left(\frac{1}{m_{{}_i}\left(x_i\right)}\right){\partial }_{x_i}{V}_{{}_I}\left(\vec{r}\right)=0;{\ddot{x}}_{{}_j}=\frac{d^2{x}_{{}_j}}{d{t}^2},$$

$${\ddot{x}}_{{}_i}+\left(\frac{\dot{m}\left(\vec{r}\right)}{m\left(\vec{r}\right)}\right){\dot{x}}_i-\frac{1}{2}\left(\frac{{\partial }_{x_i}m\left(\vec{r}\right)}{m\left(\vec{r}\right)}\right)\sum _{j=1}^n{\dot{x}}_{{}_j}^2+\left(\frac{1}{m\left(\vec{r}\right)}\right){\partial }_{x_i}{V}_{{}_{II}}\left(\vec{r}\right)=0,$$

$$\frac{d}{d\tau }\left(\frac{\partial L}{\partial {\tilde{q}}_{{}_i}}\right)-\frac{\partial L}{\partial q_{{}_i}}=0⟺\frac{d}{d\tau }{\tilde{q}}_{{}_i}+\frac{\partial }{\partial q_{{}_i}}V(\vec{q})=0.$$

$$d\tau =f\left(\vec{r}\right)dt,d{q}_{{}_i}={\delta }_{ij}\sqrt{g\left(\vec{r}\right)}d{x}_{{}_j}⟹\frac{\partial q_{{}_i}}{\partial x_{{}_j}}={\delta }_{ij}\sqrt{g\left(\vec{r}\right)}.$$

$${\hat{q}}_{{}_i}=\frac{\sum _{k=1}^n\left(\frac{\partial x_{{}_k}}{\partial q_{{}_i}}\right){\hat{x}}_k}{\sqrt{\sum _{k=1}^n{\left(\frac{\partial x_{{}_k}}{\partial q_{{}_i}}\right)}^2}}⟹{\hat{q}}_{{}_i}={\hat{x}}_{{}_i};i=1,2,\cdots ,n.$$

$${\tilde{q}}_{{}_j}=\frac{\sqrt{g\left(\vec{r}\right)}}{f\left(\vec{r}\right)}{\dot{x}}_{{}_j}⟹\frac{d}{d\tau }{\tilde{q}}_{{}_j}=\frac{\sqrt{g\left(\vec{r}\right)}}{f{\left(\vec{r}\right)}^2}\left({\ddot{x}}_{{}_j}+{\dot{x}}_{{}_j}\left[\frac{\dot{g}\left(\vec{r}\right)}{2g\left(\vec{r}\right)}-\frac{\dot{f}\left(\vec{r}\right)}{f\left(\vec{r}\right)}\right]\right),$$

$${\ddot{x}}_{{}_j}+\left(\frac{\dot{g}\left(\vec{r}\right)}{2g\left(\vec{r}\right)}-\frac{\dot{f}\left(\vec{r}\right)}{f\left(\vec{r}\right)}\right){\dot{x}}_{{}_j}+\left(\frac{f{\left(\vec{r}\right)}^2}{g\left(\vec{r}\right)}\right){\partial }_{x_j}V(\vec{q}\left(\vec{r}\right))=0;j=1,2,\cdots ,n.$$

$$\frac{d}{d\tau_{{}_{i}}}\left(\frac{\partial L}{\partial\tilde{q}_{{}_{i}}}\right)-\frac{\partial L}{\partial q_{{}_{i}}}=0\Longleftrightarrow\left\{\begin{array}[]{c}\partial q_{{}_{i}}/\partial x_{{}_{i}}=\sqrt{g_{{}_{i}}\left(x_{i}\right)}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ d\tau_{{}_{i}}\,/\vskip 6.0pt plus 2.0pt minus 2.0ptdt=f_{{}_{i}}\left(x_{i}\right)\\ \,g_{{}_{i}}\left(x_{i}\right)=m_{{}_{i}}\left(x_{i}\right)f_{{}_{i}}\left(x_{i}\right)^{2}\vskip 6.0pt plus 2.0pt minus 2.0pt\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{ }V\left(\vec{r}\right)=V\left(\vec{q}\left(\vec{r}\right)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\tilde{q}_{{}_{i}}\left(x_{i}\right)=\dot{q}_{{}_{i}}\left(x_{i}\right)/f_{{}_{i}}\left(x_{i}\right)=\dot{x}_{{}_{i}}\sqrt{m_{{}_{i}}\left(x_{i}\right)}\end{array}\right\}\Longleftrightarrow\frac{d}{dt}\left(\frac{\partial L_{{}_{I}}}{\partial\dot{x}_{i}}\right)-\frac{\partial L_{{}_{I}}}{\partial x_{i}}=0.$$

$$m\left(\vec{r}\right)\sum _{i=1}^n{\ddot{x}}_{{}_i}{\hat{x}}_{{}_i}+\dot{m}\left(\vec{r}\right)\sum _{i=1}^n{\dot{x}}_{{}_i}{\hat{x}}_{{}_i}-\frac{1}{2}\sum _{i=1}^n{\partial }_{x_i}m\left(\vec{r}\right){\hat{x}}_{{}_i}\left[\sum _{j=1}^n{\dot{x}}_{{}_j}^2\right]+\sum _{i=1}^n{\hat{x}}_{{}_i}{\partial }_{x_i}{V}_{{}_{II}}\left(\vec{r}\right)=0.$$

$$m\left(\vec{r}\right)\vec{a}+{\partial }_{r}m\left(\vec{r}\right)\left[\frac{\vec{\upsilon }\left(\vec{r}\cdot \vec{\upsilon }\right)}{r}\right]-\frac{1}{2}{\partial }_{r}m\left(\vec{r}\right)\left[\frac{\vec{r}\left(\vec{\upsilon }\cdot \vec{\upsilon }\right)}{r}\right]+\vec{\nabla }{V}_{{}_{II}}\left(\vec{r}\right)=0,$$

$$\dot{m}\left(\vec{r}\right)=\sum _{k=1}^n{\partial }_{x_k}m\left(\vec{r}\right){\dot{x}}_{{}_k}=\frac{{\partial }_{r}m\left(\vec{r}\right)}{r}\sum _{k=1}^n{x}_{{}_k}{\dot{x}}_{{}_k}=\frac{{\partial }_{r}m\left(\vec{r}\right)}{r}\left(\vec{r}\cdot \vec{\upsilon }\right);r=\sqrt{\sum _{j=1}^n{x}_{{}_j}^2}.$$

$$\left(\overrightarrow{A}\cdot \overrightarrow{B}\right)\overrightarrow{A}=\left(\overrightarrow{A}\cdot \overrightarrow{A}\right)\overrightarrow{B};\forall \overrightarrow{A}\parallel \overrightarrow{B}\in {\mathbb{R}}^n$$

$$\vec{\upsilon }\left(\vec{\upsilon }\cdot \vec{r}\right)=\vec{r}\left(\vec{\upsilon }\cdot \vec{\upsilon }\right)\text{ ; }\vec{r}\parallel \vec{\upsilon },$$

$$\vec{a}+\frac{1}{2}\frac{{\partial }_{r}m\left(\vec{r}\right)}{m\left(\vec{r}\right)}\left[\frac{\vec{r}\left(\vec{\upsilon }\cdot \vec{\upsilon }\right)}{r}\right]+\frac{1}{m\left(\vec{r}\right)}\overrightarrow{\nabla }{V}_{{}_{II}}\left(\vec{r}\right)=0.$$

$$\vec{a}+\left(\frac{{\partial }_{r}g\left(\vec{r}\right)}{2g\left(\vec{r}\right)}-\frac{{\partial }_{r}f\left(\vec{r}\right)}{f\left(\vec{r}\right)}\right)\left[\frac{\vec{r}\left(\vec{\upsilon }\cdot \vec{\upsilon }\right)}{r}\right]+\left(\frac{f{\left(\vec{r}\right)}^2}{\sqrt{g\left(\vec{r}\right)}}\right){\overrightarrow{\nabla }}_{q}V(\vec{q}\left(\vec{r}\right))=0\text{ ; }{\overrightarrow{\nabla }}_q=\sum _{i=1}^n{\hat{x}}_{{}_i}{\partial }_{q_i}\text{ ; }{\hat{q}}_{{}_i}={\hat{x}}_{{}_i}.$$

$$\frac{1}{2}\frac{{\partial }_{r}m\left(\vec{r}\right)}{m\left(\vec{r}\right)}=\frac{{\partial }_{r}g\left(\vec{r}\right)}{2g\left(\vec{r}\right)}-\frac{{\partial }_{r}f\left(\vec{r}\right)}{f\left(\vec{r}\right)}⟺\frac{1}{m\left(\vec{r}\right)}=\frac{f{\left(\vec{r}\right)}^2}{g\left(\vec{r}\right)}⟺g\left(\vec{r}\right)=m\left(\vec{r}\right)f{\left(\vec{r}\right)}^2.$$

$$\vec{a}+\frac{1}{2}\frac{{\partial }_{r}m\left(\vec{r}\right)}{m\left(\vec{r}\right)}\left[\frac{\vec{r}\left(\vec{\upsilon }\cdot \vec{\upsilon }\right)}{r}\right]+\frac{1}{m\left(\vec{r}\right)}{\overrightarrow{\nabla }}_{q}V(\vec{q}\left(\vec{r}\right))=0,$$

$$\overrightarrow{\nabla }{V}_{II}\left(\vec{r}\right)=\sqrt{g\left(\vec{r}\right)}{\overrightarrow{\nabla }}_{q}V(\vec{q}\left(\vec{r}\right))⟺\sum _{i=1}^n{\hat{x}}_{{}_i}{\partial }_{x_i}{V}_{{}_{II}}\left(\vec{r}\right)=\sum _{i=1}^n{\hat{x}}_{{}_i}{\partial }_{x_i}V(\vec{q}\left(\vec{r}\right))⟺V_{{}_{II}}\left(\vec{r}\right)=V(\vec{q}\left(\vec{r}\right)).$$

$${\ddot{x}}_{{}_i}+\left(\frac{\dot{m}\left(\vec{r}\right)}{2m\left(\vec{r}\right)}\right){\dot{x}}_i+\left(\frac{1}{m\left(\vec{r}\right)}\right){\partial }_{x_i}{V}_{{}_{II}}\left(\vec{r}\right)=0;i=1,2,\cdots ,n\text{,}$$

$${\ddot{x}}_{{}_i}+\frac{{\partial }_{r}m\left(\vec{r}\right)}{2rm\left(\vec{r}\right)}\left(\sum _{j=1}^n{\dot{x}}_{{}_j}^2\right)x_i+\frac{1}{m\left(\vec{r}\right)}{\partial }_{x_i}{V}_{{}_{II}}\left(\vec{r}\right)=0;i=1,2,\cdots ,n.$$

$$\frac{d}{d\tau}\left(\frac{\partial L}{\partial\tilde{q}_{{}_{i}}}\right)-\frac{\partial L}{\partial q_{{}_{i}}}=0\Longleftrightarrow\left\{\begin{array}[]{c}\hat{q}_{{}_{i}}=\,\hat{x}_{{}_{i}}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \partial q_{{}_{i}}/\partial x_{{}_{i}}=\sqrt{g\left(\vec{r}\right)}\,\text{\ \vskip 6.0pt plus 2.0pt minus 2.0pt}\\ d\tau\,=f\left(\vec{r}\right)\,dt\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \,g\left(\vec{r}\right)=m\left(\vec{r}\right)f\left(\vec{r}\right)^{2}\vskip 6.0pt plus 2.0pt minus 2.0pt\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{ }V\left(\vec{r}\right)=V\left(\vec{q}\left(\vec{r}\right)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \tilde{q}_{{}_{i}}\left(\vec{r}\right)=\dot{q}_{{}_{i}}\left(\vec{r}\right)/f\left(\vec{r}\right)=\dot{x}_{{}_{i}}\sqrt{m\left(\vec{r}\right)}\vskip 6.0pt plus 2.0pt minus 2.0pt\end{array}\right\}\Longleftrightarrow\frac{d}{dt}\left(\frac{\partial L_{{}_{II}}}{\partial\dot{x}_{i}}\right)-\frac{\partial L_{{}_{II}}}{\partial x_{i}}=0.$$

$$V\left(\vec{q}\right)\text{vskip}6.0ptplus2.0ptminus2.0pt=\frac{1}{2}{m}_{\circ }{\omega }_{\circ }^2\sum _{j=1}^n{q}_{{}_j}^2=\frac{1}{2}{m}_{\circ }{\omega }_{\circ }^2\left(\vec{q}\cdot \vec{q}\right).$$

$$\frac{d}{d\tau }{\tilde{q}}_{{}_i}+{\omega }_{\circ }^2{q}_{{}_i}=0;i=1,2,\cdots ,n,$$

$$q_{{}_i}=B_{{}_i}\cos \left({\omega }_{\circ }\tau +{\phi }_{{}_i}\right).$$

$$\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\vec{r},$$

Full text

**PACS numbers: **05.45.-a, 03.50.Kk, 03.65.-w

Keywords: $n$-dimensional position-dependent mass Lagrangians, nonlocal point transformation, Euler-Lagrange equations invariance, Newtonian invariance amendment, linearizability and exact solvability of PDM nonlinear Euler-Lagrange oscillators’ equations.

I Introduction

Classical and quantum mechanical particles endowed with position-dependent mass (PDM) have initiated a substantial amount of research interest over the last few decades 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ; 34 ; 35 ; 36 ; 37 ; 38 ; 39 ; 40 ; 41 ; 42 ; 43 ; 44 ; 45 ; 46 . The position-dependent mass concept is, basically, a consequential manifestation of either a position-dependent deformation in the standard constant mass setting, or a position-dependent deformation in the coordinates settings. A point mass moving within the curved coordinates/space transforms, effectively, into a position-dependent mass in Euclidean coordinates/space (c.f., e.g., 3 ; 5 ; 20 ; 45 and references cited therein). Instantaneous Galilean invariance is used to derive the PDM-Hamiltonian [46]. This would, in turn, offer a mathematically challenging problem in both classical and quantum mechanics. In quantum mechanics for example, the ordering ambiguity associated with the non-unique representation of the PDM von Roos Hamiltonian 1 possess a significant amount of arguments as to what are the most proper parametric ordering settings (e.g., 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ). It has been only recently that a proper definition for the position-dependent mass momentum operator is introduced by Mustafa and Algadhi 5 , resolving, hereby, the ordering ambiguity conflict. In classical mechanics, nevertheless, exact solutions to multidimensional PDM Euler-Lagrange equations are hard to find (e.g., 18 ; 19 ; 20 ; 21 ; 30 ; 33 ; 38 ; 44 and references cited therein). One should, therefore seek some kind of nonlocal space-time point transformations that guarantees Euler-Lagrange invariance and facilitates exact solvability.

Based on the readily existing one-dimensional version 38 ,  Mustafa in 44 has very recently embarked upon the $n$-dimensional extension of the PDM Lagrangians via a nonlocal space-time point transformation and sought invariance between the standard ”constant” mass and PDM Euler-Lagrange equations. Two $n$-dimensional PDM Lagrangian models were used,

[TABLE]

and

[TABLE]

where

[TABLE]

Herewith, $m_{\circ}$ is the standard ”constant” mass, $m_{{}_{j}}\left(x_{{}_{j}}\right)$ in $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ is a dimensionless scalar multiplier that deforms each coordinate $x_{{}_{j}}$ and/or velocity component $\dot{x}_{{}_{j}}$ in a specific functional form, and $m\left(\vec{r}\right)$ in $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ represents a common dimensionless scalar multiplier that deforms the coordinates $x_{{}_{j}}$’s and/or velocity components $\dot{x}_{{}_{j}}$’s. Hence, similar consequential position-dependent deformations in the potential force fields $V_{{}_{I}}\left(\vec{r}\right)$ and $V_{{}_{II}}\left(\vec{r}\right)$ are unavoidable in the process. Moreover, Mustafa 44 has considered a conventional constant-mass Lagrangian

[TABLE]

where $\vec{q}=\left(q_{{}_{1}},q_{{}_{2}},\cdots,q_{{}_{n}}\right)$ are some generalized coordinates and $\tau$ is a re-scaled time. The idea is simply a manifestation of Euler-Lagrange textbook invariance procedure. That is, the Euler-Lagrange equations for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ and $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ should be invariant with those for $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$, if the used nonlocal space-time point transformation is deemed useful.

However, it turned out that whilst the Euler-Lagrange equations for $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ failed to satisfy invariance conditions for $n\geq 2$, the Euler-Lagrange equations for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ proved to satisfy the invariance conditions for $n\geq 1$. Such results would consequently render the $n$-dimensional extension of the used nonlocal point transformation 44 for the $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ as a minor and/or trivial progress. The said approach 44 may very well copy and paste the very recent work for the one-dimensional PDM-Lagrangians of Mustafa 38 (along with all examples discussed and reported therein) for each degree of freedom $x_{i}$. For more details on this issue the readers may refer to Mustafa 44 . In the current methodical proposal, therefore, we propose a remedy to this invariance problem in the form of the what, hereinafter, should be called ”Newtonian invariance amendment”. This is to be viewed as a comeback of the traditional textbook Euler-Lagrange and Newtonian dynamical correspondence under PDM-settings. To the best of our knowledge, this has never been reported elsewhere in the literature. The organization of our article is in order.

In section 2, we recycle, in short, the Euler-Lagrange equations invariance for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ and $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ with those for $L\left(\vec{q},\overrightarrow{\tilde{q}};\tau\right).$ This would make the current methodical proposal self-contained and vividly instructive. In the same section, we introduce our Newtonian invariance amendment. Hereby, we show that while the textbook Euler-Lagrange invariance (for $n\geq 2$) proved satisfactory only for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$, the Newtonian invariance amendment is found to be satisfactory for both $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ and $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$, $n\geq 1$ (when compared with the conventional constant-mass Euler-Lagrange equations for $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$, of course). We may, therefore, label $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ and $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ as ”target-Lagrangians” and $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$ as ”reference Lagrangian”. In fact, the total vector presentation of Newtonian dynamical equation is nothings but the sum of all Euler-Lagrange one-coordinate equations of motion, each of which is multiplied by the corresponding unit vector (see (5), (6), (13) and (14) below). Therefore, Newtonian invariance and Euler-Lagrange invariance do not conflict each other. We devote section 3 to illustrate our methodical proposal and use some conventional constant-mass $n$-dimensional oscillators Lagrangian $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$ as a reference Lagrangian. Where, three different PDM-settings are successfully used within our nonlocal space-time point transformation recipe. They are, $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{r}$, $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{\zeta}$  (where $\,\vec{\zeta}$ is a constant vector), and $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\left(\vec{r}+\vec{\zeta}\right)$. Such vector transformational recipes are mandatory substitutional settings for our Newtonian invariance amendment. We also show that the nonlinear PDM Euler-Lagrange equations are linearizable through some nonlocal point transformations. In the same section, moreover, some $n$-dimensional illustrative examples are used. Amongst are, the Mathews-Lakshmanan type-I nonlinear PDM-oscillators of (36), the power-law type-I nonlinear PDM-oscillators (40), the Mathews-Lakshmanan type-II nonlinear PDM-oscillators (49), the power-law type-II nonlinear PDM-oscillators (51), and some nonlinear shifted Mathews-Lakshmanan type-I nonlinear PDM-oscillators (60). We conclude in section 4.

II Newtonian invariance amendment to PDM Euler-Lagrange equations

We start with recollecting/recycling some vital parts of the $n$-dimensional extension of the PDM Lagrangians (via a nonlocal point transformation) work by Mustafa 44 . Therefore, we begin with the implementation of Euler-Lagrange equations

[TABLE]

to obtain (with $m_{\circ}=1$ throughout) $n$ PDM Euler-Lagrange equations ( PDM EL-I)

[TABLE]

for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$, and PDM EL-II

[TABLE]

for $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$, where $\partial_{x_{i}}=\partial/\partial x_{i}$. Yet, the Euler-Lagrange equations for $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$ ( EL-G) yield

[TABLE]

At this point, we shall seek some sort of feasible invariance for PDM EL-I of (5) and for PDM EL-II of (6) with EL-G of (7). In so doing, Mustafa 44 has suggested that we may extend/generalize the one-dimensional nonlocal point transformation 38 to fit into the $n$-dimensional settings and re-scale both space and time through

[TABLE]

This would necessarily mean that the unit vectors in the direction of $q_{{}_{i}}$ are obtained as

[TABLE]

Where the dimensionless functional structures of $f\left(\vec{r}\right)$ and $g\left(\vec{r}\right)$ shall be determined in the process below.

Under such settings, one obtains

[TABLE]

and hence EL-G of (7) would read

[TABLE]

Obviously, the invariance between EL-I of (5) and the resulting El-G of (11) is feasible and is simply summarized by the relations

[TABLE]

However, it is clear that the dynamics of the $n$-dimensional PDM-system of $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ in (1) fully decouples and collapses into $n$ one-dimensional dynamical systems for each degree of freedom $x_{i}$ (i.e., $n$ one-dimensional PDM EL-I equations of motion). This would, in turn, render the $n$-dimensional extension proposal of Mustafa 44 as a minor and/or a trivial progress. For this approach, of 44 , may very well copy and paste our very recent work for the one-dimensional PDM-Lagrangians in 38 (along with all examples discussed and reported therein) for each degree of freedom $x_{i}$. Whereas, the comparison between El-G of (11) and PDM EL-II of (6) is only possible for the one-dimensional problems (i.e., for $n=1$). In this case, (6) collapses into (5) for $i=1=n$. Nevertheless. for the multidimensional case $n\geq 2$, the third term in (6) has no counterpart in (11). This would, in effect, make the comparison incomplete/impossible and insecure. That is, for $n\geq 2$ the Euler-Lagrange equations (6) and (11) suggest that the invariance is, apparently, still far beyond reach. Hereby, ”Newtonian invariance amendment” comes into action.

Apriori, however, one should be reminded that a transition from Euler-Lagrange into Newtonian dynamics is a simple textbook procedure and is in order. Let us recollect the PDM EL-II of (6) and rephrase it to fit into Newtonian vector dynamics. We do so by associating with each degree of freedom a corresponding unit vector $\hat{x}_{{}_{i}}$ and sum over $i=1,2,\cdots,n$ to get

[TABLE]

This would allow us to present the current equation in vector format settings, with $\partial_{r}m\left(\vec{r}\right)=\partial m\left(\vec{r}\right)/\partial r$, as

[TABLE]

where, we have used

[TABLE]

Obviously, equation (14) reduces to $m_{\circ}\,\vec{a}=-\,\,\vec{\nabla}V_{{}_{II}}\left(\vec{r}\right)$ for conventional constant mass settings (i.e., for $m_{\circ}\neq 1$ and $m\left(\vec{r}\right)=1$). At this point, nevertheless, let us consider two $n$-dimensional vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ (i.e., $\overrightarrow{A},\overrightarrow{B}\in\mathbb{R}^{n}$, where $\mathbb{R}^{n}$ is the $n$-dimensional vector space). One may assert that, if $\overrightarrow{A}$ and $\overrightarrow{B}$ are parallel to each other (i.e., co-directional $\overrightarrow{A}\parallel\overrightarrow{B}$) then $\overrightarrow{A}=\left|\overrightarrow{A}\right|\,\hat{A}=\left|\overrightarrow{A}\right|\hat{B}$ and $\overrightarrow{B}=\left|\overrightarrow{B}\right|\,\hat{B}=\left|\overrightarrow{B}\right|\,\hat{A}$. Consequently, the identity

[TABLE]

is intuitively most obvious and evident. This result would suggest that for the case where $\vec{\upsilon}$ is parallel to $\vec{r}$ one obtains

[TABLE]

to yield

[TABLE]

This would be acceptable for the case where no rotational effects are involved (i.e., the case we are considering here), otherwise the Lagrangian structure would include, in addition to the translational kinetic energy term of (2), a rotational kinetic energy term (c.f., e.g., the two-dimensional nonlinear oscillator kinetic energy term in 29 and equation (32) in 20 ) and a different treatment would be required, therefore. Similarly, in a straightforward manner, one can show that EL-G of (11) can be rewritten (in the Newtonian vector form) as

[TABLE]

We got now consistency and exact correspondence between (18) and (19). That is, the invariance between EL-G of (11) and PDM EL-II of (6) is now secured and mandates that

[TABLE]

Consequently, not only we have consistency between (18) and (19) but also we have secured Newtonian invariance between (6) and (7). We may, therefore, safely rewrite (19) as

[TABLE]

which immediately implies that $V_{{}_{II}}\left(\vec{r}\right)=V(\vec{q}\left(\vec{r}\right))$. Wherein, we have used the relations

[TABLE]

This result is to be used to determine $q_{{}_{i}}\left(\vec{r}\right)$’s as well as the form of $f(\vec{r})$ (consequently $g\left(\vec{r}\right)$) for a given $m\left(\vec{r}\right)$. Moreover, in a straightforward manner, the same procedure can be followed to show that the PDM EL-I is also Newtonian invariant. Yet, we are now able to dismantle (21) into $n$ component equations

[TABLE]

or

[TABLE]

Where both forms hold true under our current settings. As such, our nonlocal point transformation within our *”Newtonian invariance amendment” * is summarized by

[TABLE]

We may now conclude that, whilst the conventional textbook Euler-Lagrange invariance could only address the PDM EL-I settings (documented, in short, above and in a sufficiently comprehensive details in 44 ) it could not address the current PDM EL-II settings. Whereas, the what should be called, hereinafter, *”Newtonian invariance amendment” * works to perfection for both PDM EL-I and PDM EL-II settings. One should also be reminded that ”Newtonian invariance amendment” is nothings but a manifestation of the conventional constant mass Euler-Lagrange equations transition into the Newtonian vector presentation of the equation of motion. However, the Newtonian invariance amendment offered a vivid invariance to what is seemed to be incomplete and/or insecure invariance of the PDM Euler-Lagrange equations. In the forthcoming illustrative examples, we clarify our methodical proposal reported above.

III Nonlinear $n$-dimensional PDM-oscillators: Linearizability and

exact solvability

Let us start with a constant mass $m_{\circ}$ textbook harmonic oscillator force field, in the generalized coordinates,

[TABLE]

For which, one may use the EL-G equations of (7) to yield (with $m_{\circ}=1$) $n$ EL-G linear equations of motion

[TABLE]

that admit exact solutions in the form of

[TABLE]

This is going to be our reference case for the forthcoming target PDM Lagrangians. In what follows, we shall seek forms of nonlocal point transformations for $\vec{q}\left(\vec{r}\right)$ so that the resulting nonlinear PDM Euler-Largrange equations admit linearizability into (27). This would consequently suggest that one may use the exact solutions (28) of (27) to extract exact solutions for the PDM- dynamical systems of (23) or (24).

III.1 Nonlinear $n$-dimensional PDM-oscillators: $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{r}$

The substitutions of

[TABLE]

in (26), would imply the $n$-dimensional PDM-oscillators potential in the form of

[TABLE]

Before we proceed any further, we need first to put the substitution (29) to the test and see whether it satisfies our nonlocal point transformation condition $\dot{q}_{{}_{i}}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}f\left(\vec{r}\right)\dot{x}_{{}_{i}}$, of (25), or not. This is done in order.

[TABLE]

We may now swap $\vec{r}$ and $\vec{\upsilon}$ in (17) and rewrite (31) as

[TABLE]

Comparing this result with $\dot{q}_{{}_{i}}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}f\left(\vec{r}\right)\dot{x}_{{}_{i}}$ of (25), we obtain

[TABLE]

We, therefore, conclude that the substitution (29) satisfies our Newtonian invariance amendment provided that $f\left(\vec{r}\right)$ is given by (33). As such, equations (24) read

[TABLE]

which are our target PDM Euler-Lagrange equations to be solved for different PDM settings. Obviously, a reverse engineering of (34) into (27), through the nonlocal transformation (25), would institute a linearization process of (34) into (27). Under such nonlocal point transformation, we consider two $n$-dimensional PDM-oscillator examples.

III.1.1 Mathews-Lakshmanan type-I $n$-dimensional PDM-oscillators: $m\left(\vec{r}\right)=1/\left(1\pm\lambda r^{2}\right)$

With a PDM function $m\left(\vec{r}\right)=1/\left(1\pm\lambda r^{2}\right)$, one can show that

[TABLE]

Under such settings, the PDM Euler-Lagrange equations of (34) imply the Mathews-Lakshmanan type-I $n$-dimensional PDM-oscillators equations of motion

[TABLE]

Which admit linearizability through the nonlocal transformation (25) and inherit exact solutions of the form

[TABLE]

As such, the total energy is given by

[TABLE]

III.1.2 $n$-dimensional PDM power–law type-I oscillators: $m\left(\vec{r}\right)=k\,r^{2\upsilon}$

A power-law type PDM function $m\left(\vec{r}\right)=k\,r^{2\upsilon}$ would imply that

[TABLE]

Hence, the PDM Euler-Lagrange equations of (34) yield the $n$-dimensional PDM-oscillators equations of motion

[TABLE]

Such nonlinear equations of motion admit linearizability through the nonlocal transformation (25) and inherit exact solutions of the form

[TABLE]

where $\upsilon\neq-1,0$, otherwise trivial solutions are manifested. Moreover, the total energy reads

[TABLE]

III.2 Nonlinear $n$-dimensional PDM-oscillators: $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{\zeta}$

Let us now use the assumption that

[TABLE]

in (26), would imply the $n$-dimensional PDM-oscillators potential

[TABLE]

Where $\vec{\zeta}$ is a constant vector and is parallel to $\vec{r}\,$ and $\vec{\upsilon}$ (i.e., $\vec{r}\,\parallel\vec{\upsilon}\parallel\vec{\zeta}$ ) and satisfies the vector identity in (17). Under such settings, one would obtain

[TABLE]

which immediately implies that

[TABLE]

This result is consistent with $\dot{q}_{{}_{i}}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}f\left(\vec{r}\right)\dot{x}_{{}_{i}}$ of (25) provided that

[TABLE]

Therefore, the PDM Euler-Lagrange equations of (24) read

[TABLE]

This result represents now our new target PDM Euler-Lagrange equation to be solved for different PDM settings. Yet again, a reverse engineering of (48) into (27), through the nonlocal transformation (25), would institute a linearization process of (48) into (27). Under such nonlocal point transformation, we consider two $n$-dimensional PDM-oscillator examples.

III.2.1 Mathews-Lakshmanan type-II $n$-dimensional PDM-oscillators: $m\left(\vec{r}\right)=1/\left(1\pm\lambda r^{2}\right)$

With the substitution

[TABLE]

the PDM Euler-Lagrange equations of (48) now read

[TABLE]

Which is exactly the same as (36) provided that $\zeta^{2}=\mp 1/\lambda$ (hence the notion ”Mathews-Lakshmanan type-II $n$-dimensional PDM-oscillators” is deemed appropriate). Such nonlinear equations of motion admit linearizability through the nonlocal transformation (25) and inherit the exact solutions of of (37) and (38).

III.2.2 $n$-dimensional PDM power–law type-II oscillators: $m\left(\vec{r}\right)=\lambda r^{2\upsilon}$

A power-law type PDM function $m\left(\vec{r}\right)=\lambda r^{2\upsilon}$ would imply that

[TABLE]

and consequently the PDM Euler-Lagrange equations of (48) read

[TABLE]

Such nonlinear equations of motion are linearizable into (27), through the nonlocal transformation (25), and admit exact solutions in the form

[TABLE]

provided that $\upsilon=-1$, and $\lambda=-1/\xi^{2}$. Hence, the total energy reads

[TABLE]

III.3 Nonlinear $n$-dimensional shifted PDM-oscillators: $\vec{q}\left(\vec{y}\right)=\sqrt{m\left(y\right)}\,\vec{y},\,\vec{y}=\left(\vec{r}+\vec{\zeta}\right)$

A mixture of the two cases above, $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{r}$ and $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{\zeta}$ would imply that

[TABLE]

In this case, the oscillator potential of (26) yields an $n$-dimensional shifted PDM oscillator potential

[TABLE]

Under such shifted PDM settings, one may rewrite the transformation recipe (25) as

[TABLE]

Which, consequently, suggests that the $n$-dimensional PDM Euler-Lagrange equations are

[TABLE]

where

[TABLE]

It is obvious that for a PDM function of the form

[TABLE]

equations (57) would read the what may, very well, be called the shifted Mathews-Lakshmanan type-I $n$-dimensional PDM-oscillators

[TABLE]

Such nonlinear equations of motion are linearizable into (27), through the nonlocal transformation (56), and admit exact solutions in the form

[TABLE]

The total energy is then

[TABLE]

which is in an obvious resemblance as that in (38).

IV Concluding Remarks

In this work, we have considered two-types of PDM-Lagrangians $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ of (1) and $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ of (2). They represent two different PDM-Lagrangians structures. In $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$, $m_{{}_{j}}\left(x_{{}_{j}}\right)$ is a dimensionless scalar deformation in the coordinate $x_{{}_{j}}$ and/or velocity component $\dot{x}_{{}_{j}}$ in a specific functional form. Whereas, in $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$, $m\left(\vec{r}\right)$ is a common dimensionless deformation for all coordinates $x_{{}_{j}}$ and/or velocity components $\dot{x}_{{}_{j}}$. The feasibility of their textbook Euler-Lagrange invariance with the conventional constant mass Lagrangians $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$ of (3) is studies via some $n$-dimensional nonlocal space-time point transformation recipe of (8), (9), and (10). We have shown that, while the PDM Euler-Lagrange equations for $L_{{}_{I}}\left(\vec{r},\vec{\upsilon};t\right)$ satisfy the invariance conditions with EL-G of $L\left(\overrightarrow{q},\overrightarrow{\tilde{q}};\tau\right)$ for $n\geq 1$, the PDM Euler-Lagrange equations for $L_{{}_{II}}\left(\vec{r},\vec{\upsilon};t\right)$ failed to do so for $n\geq 2$. This issue has stimulated and/or inspired the current methodical proposal to introduce the new concept of ”Newtonian invariance amendment”.

As long as the conventional constant mass setting are in point, both Euler-Lagrange invariance and *Newtonian invariance *coincide with each other. However, under the current PDM $n$-dimensional setting, it is deemed necessary and vital that the transition from the Euler-Lagrange component presentations, (6) and (11), to Newtonian vector presentations, (18) and (19), should be carried out in order to secure invariance. It was obvious that for $n\geq 2$ the invariance between Euler-Lagrange equations (6) and (11) is still far beyond reach. Whereas, in the Newtonian presentations the invariance between (18) and (19) is proved crystal clear. The totality of the Newtonian vector presentation of the dynamical equation of motion is shown to be more comprehensive/instructive than the Euler-Lagrange components presentations. Hence, the notion ”Newtonian invariance amendment” is rendered unavoidable for the current methodical proposal. Yet, the Newtonian invariance amendment has offered a vivid invariance to what is seemed to be incomplete and/or insecure invariance of the PDM Euler-Lagrange equations. This is clarified and documented in our analytical discussions in section 2. To the best of our knowledge, this issue has never been reported elsewhere in the literature.

The $n$-dimensional linear Euler-Lagrange equations of motion (27) for the oscillators Lagrangian

[TABLE]

of (3) and (26), is used in section 3. Therein, we have used the substitutions $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{r}$, $\vec{q}\left(\vec{r}\right)=\sqrt{m\left(\vec{r}\right)}\,\vec{\zeta}$, and $\vec{q}\left(\vec{y}\right)=\sqrt{m\left(\vec{y}\right)}\,\vec{y}$   (where $\vec{y}=\sum\limits_{k=1}^{n}\left(x_{{}_{k}}+\xi_{{}_{k}}\right)\hat{x}_{{}_{k}}$) to find the corresponding $n$-dimensional nonlinear PDM-oscillators’ Euler-Lagrange equations of motion for the Lagrangians

[TABLE]

[TABLE]

and

[TABLE]

Their Euler-Lagrange equations (34), (48), and (57), respectively, satisfy the Newtonian invariance amendment. Their linearizability into (27) through some nonlocal transformations, (25) and (56), is discussed for different PDM settings. We have used some illustrative examples that include, the* Mathews-Lakshmanan type-I PDM-oscillators* (36), the power-law type-I PDM-oscillators (40), the Mathews-Lakshmanan type-II PDM-oscillators (49), the power-law type-II PDM-oscillators (51), and some nonlinear shifted Mathews-Lakshmanan type-I PDM-oscillators (60). Hereby, using the exact solutions (28) of the standard $n$-dimensional simple harmonic oscillator (27), we have successfully extracted exact solutions for the PDM-dynamical systems in (36), (40), (49), (51), and (60).

Finally, we have not only introduced the new concept of the Newtonian invariance amendment as an alternative to Euler-Lagrange invariance, but also we foresee that a new class of * pseudo-superintegrable* and/or pseudo-superseparable PDM-Lagrangians (and consequently PDM-Hamiltonians) is implicitly introduced in the current methodical proposal. The Lagrangian in (63), hence the corresponding Hamiltonian, represent a class of *superintegrable *Lagrangians/Hamiltonians in the Liouville-Arnold sense of integrability (for more details on this issue the reader may refer to the sample of references 39 ; 40 ; 41 ; 42 ; 43 ; 44 ; 45 and related references cited therein). That is, they introduce more constants of motion (also called integrals of motion) than the degrees of freedom the system is moving within. As long as the  *superintegrable *Lagrangian (63), and its corresponding Hamiltonian, are transformable (through the current nonlocal point transformation) into a set of PDM-Lagrangians/Hamiltonians that do not even admit separability or integrability, the descendent PDM-Lagrangians/Hamiltonians deserve to be labeled as *pseudo-superintegrable *and/or pseudo-superseparable, therefore.