Modified Emden and the corresponding standard PDM dynamical equations: exact solvability and classical-states crossings
We introduce the position-dependent mass (PDM) settings for the modified Emden equation (MEE). To work out exact solutions for the PDM-MEE, we use the most general solution for the simple harmonic oscillator's dynamical equation (i.e., the initial conditions shall be identified by the PDM-MEE problem at hand) and report the most general solution for the MEE. In so doing, we use a nonlocal point transformation. Next, we map MEE-solution into the corresponding PDM-MEE. Two PDM-MEE-type classical particles are used as illustrative examples, and their exact solutions are reported. Under specific parametric considerations, the phase-space trajectories are observed to have classical-states crossings as they evolve in time (where each phase-space trajectory represents a classical state). This observation holds true not only for PDM-MEE-type classical particles but also for MEE-type particles.