  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.
