Notes on the central forcern
Notes on the central forcern
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关于中央力的注意事项
DOI：
10.1007/bf00642162
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发表时间：
1980
影响因子：
1.9
通讯作者：
R. Broucke
中科院分区：
文献类型：
--
作者：
R. Broucke
在这篇文章中，我们收集了几个结果有关的经典问题的二维运动的一个粒子在一个领域的中心力成正比的真实的权力的距离。首先推广了Whittaker关于r的14次幂的结果，使之与椭圆函数可积。我们列举了六种更一般的势函数，包括Whittaker的十四种势函数作为特例（第2和3节）。接下来，我们研究了Whittaker术语中的圆形解的稳定性，即问题的奇异解。稳定性指数作为指数的函数计算，其性质进行了解释，特别是在分叉与其他家庭的普通周期解（第4，5和7节）。在第六节中，我们利用一个类似于Kepler问题的偏心反常的辅助变量，给出了立方体力反问题的详细解，最后证明了有心力问题的稳定奇异圆解推广到两定心问题的稳定奇异椭圆解。同时也讨论了两固定中心问题的稳定性和与其它周期解族的分歧。
In this article we collect several results related to the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. At first we generalize Whittaker's result of the fourteen powers ofrwhich lead to intergrability with elliptic functions. We enumerate six more general potentials, including Whittaker's fourteen potentials as particular cases (Sections 2 and 3).Next, we study the stability of the circular solutions, which are the singular solutions of the problem, in Whittaker's terminology. The stability index is computed as a function of the exponentnand its properties are explained, especially in terms of bifurcations with other families of ordinary periodic solutions (Sections 4, 5 and 7). In Section 6, the detailed solution of the inverse cube force problem is given in terms of an auxiliary variable which is similar to the eccentric anomaly of the Kepler problem.Finally, it is shown that the stable singular circular solutions of the central force problem generalize to stable singular elliptic solutions of the two-fixed-center problem. The stability and the bifurcations with other families of periodic solutions of the two-fixed-center problem are also described.