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We argue that, under multidimensional position-dependent mass (PDM) settings,the Euler-Lagrange textbook invariance falls short and turned out to be vividlyincomplete and/or insecure for a set of PDM-Lagrangians. We show that thetransition from Euler-Lagrange component presentation to Newtonian vectorpresentation is necessary and vital to guarantee invariance. The totality ofthe Newtonian vector equations of motion is shown to be more comprehensive andinstructive than the Euler-Lagrange component equations of motion (they do notrun into conflict with each other though). We have successfully used theNewtonian invariance amendment, along with some nonlocal space-time pointtransformation recipe, to extract exact solutions for a set of n-dimensionalnonlinear PDM-oscillators. They are, Mathews-Lakshmanan type-I PDM-oscillators,power-law type-I PDM-oscillators, the Mathews-Lakshmanan type-IIPDM-oscillators, the power-law type-II PDM-oscillators, and some nonlinearshifted Mathews-Lakshmanan type-I PDM-oscillators.

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