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.
Categories: nlin.SI