论文标题
一个新型潜力，带有中心圆形轨道
A Novel Potential Featuring Off-Center Circular Orbits
论文作者
论文摘要
在第1本书中，命题7，他的1687年哲学哲学哲学prinistia Mathematica的问题2中，艾萨克·牛顿（Isaac Newton）提出并回答了以下问题：让粒子在中央力量领域移动的轨道是一个偏离中心的圆圈。力的大小如何取决于粒子的位置？在本文中，我们确定只能在零能量下产生这种力的潜力。我们进一步绘制了零能源轨道的潜力，以在球体上有限能源轨道。这种双重性是古尔萨特（Goursat）从1887年开始的一般结果的一个特定实例。地图本身是一个反向立体投影，这一事实解释了感兴趣系统中零能轨道的循环。最后，我们确定了运动的附加组成部分 - 库仑问题中runge -lenz载体的类似物 - 这是导致我们问题中零能轨道的接近性。
In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.
