AI 中文总结
该研究提出了开普勒第一定律的简单几何证明，基于速度投影比这一运动学不变量，揭示行星轨道的圆锥曲线性质源于平方反比定律。
AI 中文摘要
我们给出开普勒第一定律的一个简单几何证明。该论证的核心是一个我们称之为速度投影比的运动学不变量，即速度的径向投影与速度在平行于近拱点连线的固定方向上的投影之比。我们证明该比值的恒定性是牛顿定律的直接推论，且它直接导出轨道的焦点-准线性质。通过建立行星动力学与圆锥曲线经典几何定义之间的直接联系，我们的证明揭示了轨道的圆锥曲线性质是平方反比定律的必然结果。
英文摘要
We present a simple geometric proof of Kepler's first law. The core of the argument rests on a kinematic invariant that we call the velocity projection ratio - the ratio between the radial projection of the velocity and its projection onto a fixed direction parallel to the periapsis line. We show that the constancy of this ratio is an immediate consequence of Newton's laws, and that it leads directly to the focus-directrix property of the orbit. By establishing a direct link between planetary dynamics and the classical geometric definition of conics, our proof reveals the conic nature of the orbit as an inevitable consequence of the inverse-square law.
