Position-dependent mass Lagrangians: nonlocal transformations, Euler-Lagrange invariance and exact solvability

A general non-local point transformation for position-dependent mass Lagrangians and their mapping into a "constant unit-mass" Lagrangians in the generalized coordinates is introduced. The conditions on the invariance of the related Euler-Lagrange equations are reported. The harmonic oscillator linearization of the PDM Euler-Lagrange equations is discussed through some illustrative examples including harmonic oscillators, shifted harmonic oscillators, a quadratic nonlinear oscillator, and a Morse-type oscillator. The Mathews-Lakshmanan nonlinear oscillators are reproduced and some "shifted" Mathews-Lakshmanan nonlinear oscillators are reported. The mapping of an isotonic nonlinear oscillator into a PDM deformed isotonic oscillator is also discussed.