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        "description": "The stationary setting bounds positions and velocities and requires the relevant averages or periodic averages to be meaningful. A small boundary term alone does not establish existence of every time average or a general ergodic theorem. In the original convention, vis viva is one-half mass times speed squared; the uniform-pressure contribution is 3pV/2. Pair-force and molecular phase-average specializations retain their additional assumptions. This mechanical contribution differs from Clausius entropy theory and later astronomical applications. The source is a contemporary English translation, not a claim of identical German-source bytes.",
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      "statement": "# Clausius: the mechanical virial relation for bounded stationary motion\n\n## Can motion tell us about forces?\n\nImagine a particle moving back and forth on a spring. Its speed and displacement change continuously, yet over a whole cycle their averages are related. Clausius mechanical virial relation connects average kinetic energy with the positions and forces acting in a stationary system. Stationary here is a condition on the motion and its averaging behaviour; it does not mean every particle remains motionless. This is a mechanical result, distinct from his entropy formulation.\n\n## The boundary term explains the condition\n\nMultiply each particle momentum by its position and differentiate with time. The derivative contains twice the kinetic energy plus the sum of position times force. Average over a time interval: the derivative becomes an endpoint difference divided by the elapsed time. If that boundary term vanishes and the needed averages exist, twice the average kinetic energy equals the negative average of the position-force sum. Periodic bounded motion supplies a particularly easy case to check.\n\n## A spring makes the identity visible\n\nUse a teaching model with mass one, spring constant one and motion x equals cosine of time. Its velocity is minus sine of time, and its force is minus x. Over a complete cycle, sine squared and cosine squared each average one-half. Average kinetic energy is therefore one-quarter. The average of x times force is minus one-half. Twice one-quarter equals the negative of minus one-half, so the virial identity balances exactly in this example.\n\n## A finite snapshot is a different claim\n\nStop the same motion at an arbitrary time and the endpoint difference need not vanish. Bounded positions and velocities make that difference divided by a sufficiently long time small, but they do not by themselves prove every time average exists. A galaxy cluster also requires physical assumptions before an observed velocity spread can be turned into a mass estimate. That later application is valuable without becoming part of the original theorem authorship.\n\n## Leave the next reader a reproducible check\n\nAsk your agent to compare the two sides over one full spring cycle and over a shorter interval. Require it to show the endpoint correction, rather than quietly discarding a mismatch. Label the spring model as an illustration, not an experiment reported in 1870. A short derivation and an explicit failed finite-interval equality help later users understand the scope. An unexplained numerical fit or a general claim that all systems are ergodic does not.\n\n## Scientific scope\n\nThe stationary setting bounds positions and velocities and requires the relevant averages or periodic averages to be meaningful. A small boundary term alone does not establish existence of every time average or a general ergodic theorem. In the original convention, vis viva is one-half mass times speed squared; the uniform-pressure contribution is 3pV/2. Pair-force and molecular phase-average specializations retain their additional assumptions. This mechanical contribution differs from Clausius entropy theory and later astronomical applications. The source is a contemporary English translation, not a claim of identical German-source bytes.\n\n## Sources for this explanation\n\n- [On a Mechanical Theorem applicable to Heat (PhilosophicalMagazine,series4,40,122–127)](https://commons.princeton.edu/josephhenry/wp-content/uploads/sites/71/2021/01/1870-Virial-Thm-Classius-The-London-Edinburgh-and-Dublin-Philosophical-Magazine-and-Journal-of-Science.pdf): Contemporary English translation, printed pp.122–127; averaging conditions, vis viva convention, force relation and pressure specialization. German-original bytes not collated.\n",
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      {
        "title": "Can motion tell us about forces?",
        "text": "Imagine a particle moving back and forth on a spring. Its speed and displacement change continuously, yet over a whole cycle their averages are related. Clausius mechanical virial relation connects average kinetic energy with the positions and forces acting in a stationary system. Stationary here is a condition on the motion and its averaging behaviour; it does not mean every particle remains motionless. This is a mechanical result, distinct from his entropy formulation."
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        "title": "The boundary term explains the condition",
        "text": "Multiply each particle momentum by its position and differentiate with time. The derivative contains twice the kinetic energy plus the sum of position times force. Average over a time interval: the derivative becomes an endpoint difference divided by the elapsed time. If that boundary term vanishes and the needed averages exist, twice the average kinetic energy equals the negative average of the position-force sum. Periodic bounded motion supplies a particularly easy case to check."
      },
      {
        "title": "A spring makes the identity visible",
        "text": "Use a teaching model with mass one, spring constant one and motion x equals cosine of time. Its velocity is minus sine of time, and its force is minus x. Over a complete cycle, sine squared and cosine squared each average one-half. Average kinetic energy is therefore one-quarter. The average of x times force is minus one-half. Twice one-quarter equals the negative of minus one-half, so the virial identity balances exactly in this example."
      },
      {
        "title": "A finite snapshot is a different claim",
        "text": "Stop the same motion at an arbitrary time and the endpoint difference need not vanish. Bounded positions and velocities make that difference divided by a sufficiently long time small, but they do not by themselves prove every time average exists. A galaxy cluster also requires physical assumptions before an observed velocity spread can be turned into a mass estimate. That later application is valuable without becoming part of the original theorem authorship."
      },
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        "text": "Ask your agent to compare the two sides over one full spring cycle and over a shorter interval. Require it to show the endpoint correction, rather than quietly discarding a mismatch. Label the spring model as an illustration, not an experiment reported in 1870. A short derivation and an explicit failed finite-interval equality help later users understand the scope. An unexplained numerical fit or a general claim that all systems are ergodic does not."
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