Rudolf Clausius established a relation between average kinetic energy and the forces acting on a bounded mechanical system. The virial theorem follows by differentiating the sum of position-momentum products and averaging when the boundary term remains bounded. For inverse-square gravity it gives under the appropriate equilibrium assumptions. The result connects microscopic motion with binding energy and supplies a bridge between mechanics, gases and gravitating systems.[ref]
On a Mechanical Theorem applicable to Heat by R. Clausius. 1870 paper, mechanical time-average relation; gravitational specialization in modern notation
Portrait of Rudolf Clausius
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Rudolf Clausius (1822–1888) was a German physicist and mathematician best known for his contributions to the field of thermodynamics. He is particularly recognized for formulating the second law of thermodynamics and introducing the concept of entropy, which quantifies the amount of disorder or randomness in a system. Clausius's work laid the foundation for understanding the fundamental principles governing heat transfer and energy conversion.