标题： 牛顿第二定律的线性化 显示英文标题

标题： Linearization of Newton's second law

摘要： 非线性微分方程的几何线性化是一种构建解析解的稳健方法。 该方法与李对称性的存在有关，这些对称性可用于确定点变换，从而将给定的微分方程写成线性形式。 在本研究中，我们采用另一种几何方法，并利用Eisenhart提升来几何线性化描述在自主力作用下沿直线运动的粒子的牛顿系统。 我们的研究结果表明，对于振子、带有或不带振子项的Ermakov势以及Morse势，牛顿第二定律可以全局表示为自由粒子的形式。 本研究为通过等效动力系统对微分方程进行几何线性化开辟了新的方向。 显示英文摘要

摘要： The geometric linearization of nonlinear differential equation is a robust method for the construction of analytic solutions. The method is related to the existence of Lie symmetries which can be used to determine point transformations such that to write the given differential equation in a linear form. In this study we employ another geometric approach and we utilize the Eisenhart lift to geometric linearize the Newtonian system describing the motion of a particle in a line under the application of an autonomous force. Our findings reveal that for the oscillator, the Ermakov potential with or without the oscillator term, and the Morse potential, Newton's second law can be globally expressed in the form of that of a free particle. This study open new directions for the geometric linearization of differential equations via equivalent dynamical systems.