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 solvability 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, p = m (r) ṙ, but it is rather related to the time derivative of the pseudo-momentum, π (r) = m (r)ṙ (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 ≥ 2. The Euler-Lagrange invariance falls short/incomplete for n ≥ 2 under PDM settings. An alternative invariance is therefore sought in the so called, hereinafter, n-dimensional PDM Ḣ-invariance (i.e., time derivative of the Hamiltonian). Such invariance, in addition to Newtonian invariance of Mustafa [42] authorizes, in effect, 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.

Quantum solvability of quadratic Li'enard type nonlinear oscillators possessing maximal Lie point symmetries: An implication of arbitrariness of ordering parameters

“…We hope to pursue this question further. the one dimensional nonlinear oscillators which possess isochronous solutions [7]. The position dependent mass nonlinear oscillators studied in [7] are…”

“…Recently, Mustafa [7] studied the isochronicity, linearizability and exact solvability of some one dimensional and n-dimensional position dependent mass nonlinear oscillators corresponding to (A.1). In this section, we analyze the quantum solvability of some of…”

“…Classically certain Liénard type-I and type-II nonlinear oscillators were shown to possess nonstandard Hamiltonians characterized by the nonlinear parameter [1][2][3][4][5][6][7]. Under appropriate limit on the system parameters or through suitable transformations (local/ nonlocal), the Hamiltonians can be related to that of the linear harmonic oscillator.…”

“…We will consider the quantum treatment for the above two equations here. Other examples (see for example, [7]) can be treated in a similar fashion as these examples as indicated in Appendix B. In Eqs.…”