SIAM Review, Volume 65, Issue 2, Page 563-588, May 2023.
A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation. We present a comprehensive proof of Bertrand's Theorem that is accessible to undergraduate students who are familiar with basic notions from advanced calculus and differential equations.

中文翻译：

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伯特兰定理的综合证明

SIAM Review，第 65 卷，第 2 期，第 563-588 页，2023 年 5 月。
牛顿力学的一个基石成果是关于经典二体问题解行为的伯特兰定理。它指出，在所有可能的引力定律中，只有两个表现出所有有界轨道都是封闭的性质。其中之一是牛顿引力，另一个是胡克引力。我们提供了 Bertrand 定理的综合证明，熟悉高等微积分和微分方程基本概念的本科生可以使用该证明。