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Gravity is the weakest of the four fundamental forces of physics. The standard gravity theory is Newton's law of universal gravitation and general theory of relativity (proposed by Albert Einstein in 1915, and David Hilbert, and others). Alternative formulations include string theory, entanglement and others.

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Is there a high level reason why the inverse square law of gravitation yields periodic orbit...

Here is an interpretation using symmetry reduction, but without explicitly using the Lenz-Runge vector (it's essentially an extended version of the example given in Cushman & Bates "Global aspects of …
Tobias Diez's user avatar
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