On the n-dimensional extension of Position-dependent mass Lagrangians: nonlocal transformations, Euler--Lagrange invariance and exact solvability

The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa <cite>38</cite> is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa <cite>38</cite>, we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass position-dependent. Two illustrative examples are used as reference Lagrangians for different PDM settings, the nonlinear n-dimensional PDM-oscillators and the nonlinear isotonic n-dimensional PDM-oscillators. Exact solvability is also indulged in the process.