= Virial theorem derivation and history
The proof is a single <time derivative>. Apply the <product rule> to each <dot product> and use <Newton's second law>:
$$
\dot G=\sum_i\dot{\mathbf p}_i\cdot\mathbf r_i+\sum_i\mathbf p_i\cdot\dot{\mathbf r}_i
=\sum_i\mathbf F_i\cdot\mathbf r_i+2K.
$$
Integrating from $0$ to $T$ gives the endpoint identity; the stated sublinear condition removes the endpoint term only in the infinite-<time> limit. Equivalently, for the scalar inertia $I=\tfrac12\sum_i m_i|\mathbf r_i|^2$ used in the <pairwise moment-of-inertia identity>, $\dot I=G$ and $\ddot I=2K+\sum_i\mathbf r_i\cdot\mathbf F_i$. Specifying this factor of one half avoids confusion with another common inertia convention.
In <On a Mechanical Theorem applicable to Heat by R. Clausius> (1870, pp. 123-124), the author defines stationary motion through bounded <positions> and <velocities>, includes planetary <orbits> and elastic vibrations, and names the average negative half of the force-position sum the “virial”. That historical convention differs from the auxiliary $G(t)$ defined above. His proof on pp. 126-127 keeps an endpoint term, then explains why it vanishes over a complete period or becomes small in a long bounded irregular motion. The same paper separates internal and external <forces> and includes uniform boundary <pressure> on pp. 125-126: confinement was already part of the formulation, rather than a reason to omit external stresses.<On a Mechanical Theorem applicable to Heat by R. Clausius>[On a Mechanical Theorem applicable to Heat by R. Clausius (1870), pp. 123-127; p. 122 footnote dates the lecture to 13 June 1870]{ref}
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