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.

中文翻译：

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n 维 PDM 非线性振荡器：线性化和欧拉-拉格朗日或牛顿不变性

我们认为，在多维位置相关质量 (PDM) 设置下，欧拉-拉格朗日教科书的不变性不足，并且对于一组 PDM-拉格朗日函数来说显然是不完整和/或不安全的。我们表明，从欧拉-拉格朗日分量表示到牛顿向量表示的过渡对于保证不变性是必要且至关重要的。牛顿矢量运动方程的整体被证明比欧拉-拉格朗日分量运动方程更全面和更有指导意义（尽管它们不会相互冲突）。我们已经成功地使用牛顿不变性修正以及一些非局部时空点变换方法来提取一组 n 维非线性 PDM 振荡器的精确解。它们是 Mathews-Lakshmanan I 型 PDM 振荡器，