一种具有偏心圆轨道的新势

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
M. Olshanii
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引用次数: 1

摘要

牛顿在1687年出版的《自然哲学的数学原理》第一卷第七命题第二题中提出并回答了如下问题:假设在中心力场中运动的粒子的轨道是偏离中心的圆。力的大小如何取决于粒子在圆上的位置?在这篇文章中,我们确定了一个可以产生这样一个力的势能,只有在零能量的情况下。我们进一步将这个势域中的零能量轨道映射到球面上的有限能量自由运动轨道;这种对偶是古尔萨在1887年得出的一般结果的一个特例。这张图本身是一个逆立体投影,这一事实解释了我们感兴趣的系统中零能量轨道的圆度。最后,我们确定了一个额外的运动积分——类似于库仑问题中的龙格-伦茨向量——它负责我们问题中零能量轨道的接近程度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Novel Potential Featuring Off-Center Circular Orbits
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.
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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