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

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, $\mathbf{p}$, but it is rather related to the time derivative of the pseudo-momentum, $\mathbf{π}\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$ (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 $n\ge 2$. The Euler-Lagrange invariance falls short/incomplete for $n\ge 2$ 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.

15 pages and 6 figures

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• On the classical and quantum dynamics of a class of nonpolynomial oscillators
• $n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability
• PDM Klein-Gordon particles in Som-Raychaudhuri cosmic string spacetime with space-like dislocation and a non-uniform magnetic field: spacetime associated degeneracies, vorticity-energy, and charge-energy correlations
• Liénard Type Nonlinear Oscillators and Quantum Solvability
• A Class of Isochronous and Non-Isochronous Nonlinear Oscillators
• Dynamics of the Modified Emden and pseudo-Modified Emden Equations: position-dependent mass, invariance and exact solvability
• Quantum solvability of quadratic Li'enard type nonlinear oscillators possessing maximal Lie point symmetries: An implication of arbitrariness of ordering parameters