= Rudolf Clausius
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<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 $2\langle K\rangle+\langle U\rangle=0$ under the appropriate equilibrium assumptions. The result connects microscopic motion with binding <energy> and supplies a bridge between <mechanics>, gases and gravitating systems.<On a Mechanical Theorem applicable to Heat by R. Clausius>[1870 paper, mechanical time-average relation; gravitational specialization in modern notation]{ref}
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{title=Portrait of <Rudolf Clausius>}
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