Periodic orbits are the key for understanding classical Hamiltonian systems. As we show here, they are the clue for understanding Bertrand’s result relating the boundedness, flatness, and periodicity of orbits with the functional form of the potentials producing them. This result, which is known as Bertrand’s theorem, was proved in 1883 using classical 19th century techniques. In this paper, we prove such a result using the relationship between the bounded plane and periodic orbits, constants of motion, and continuous symmetries in the Hamiltonian system.

中文翻译：

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周期轨道，超可积性和贝特朗定理

周期轨道是理解经典哈密顿系统的关键。正如我们在此处所示，它们是理解贝特朗结果的线索，该结果将轨道的有界性，平坦性和周期性与产生它们的势能的功能形式联系起来。该结果被称为贝特朗定理，于1883年使用19世纪的经典技术进行了证明。在本文中，我们利用哈密顿系统中的有界平面与周期轨道，运动常数和连续对称性之间的关系证明了这种结果。