带平方阻力的牛顿运动方程和Toda势的可解方程
Newton's equation of motion with quadratic drag force and Toda's potential as a solvable one
摘要：
完全确定了一维二次阻力系统牛顿运动方程的精确可解势族。该确定是基于对任何潜在形状有效的隐式反函数解，因此是详尽的。这个可解族包括出现在Toda格中的指数势作为特殊极限。通过匹配适用于正速度和负速度的解来构造全局解，产生在三阶导数中具有尖点的分段解析函数，即，那个混蛋这些步骤和特征可以被看作是Gorder构造[Phys. Scr. 2015，{ \bf 90}，085208]中公开的方法。通过求解匹配方程，导出了系统的渐近公式，并证明了振幅阻尼与$ t^{-1} $成正比。
英文摘要：
The family of exactly solvable potentials for Newton's equation of motion in the one-dimensional system with quadratic drag force has been determined completely. The determination is based on the implicit inverse-function solution valid for any potential shape, and hence exhaustive. This solvable family includes the exponential potential appearing in the Toda lattice as a special limit. The global solution is constructed by matching the solutions applicable for positive and negative velocity, yielding the piecewise analytic function with a cusp in the third-order derivative, i.e., the jerk. These procedures and features can be regarded as a generalization of Gorder's construction [Phys. Scr. 2015, {\bf 90}, 085208] to the energy-dissipating damped oscillators. We also derive the asymptotic formulae by solving the matching equation, and prove that the damping of the oscillation amplitude is proportional to $ t^{-1} $.
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