The nonlinear generalized modified Emden-type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determines the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e. no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.

众所周知，非线性广义修正 Emden 型方程 (GMEE) 通过一些非局部变换可线性化为简谐振子 (HO) 或阻尼谐振子 (DHO)。因此，我们表明，非局部变换的结构和对 HO 或 DHO 的线性化决定了所涉及的动力的性质/结构（因此，决定了动力学方程的结构）。然而，使用逆向工程策略，以便对新兴 GMEE 的精确解进行非局部变换，以找到 HO 和 DHO 动力学方程的精确解。因此，虽然 HO 的精确解仍然是教科书，但 DHO 的精确解（从未在其他地方报道过，据我们所知）显然是最明确和最通用的解决方案，它为相关的欠阻尼、临界阻尼和过阻尼情况提供一致性和全面覆盖（即没有复杂的坐标设置和/或速度显着/可行）。此外，对于所有新兴的动力系统，我们报告了每个解决方案的说明性数字以及相应的相空间轨迹，因为它们随时间演变。