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The living force or the vis-viva is a quantity usually attributed to Leibniz (although there were a few other people who identified it as a conserved quantity in certain collisions earlier). Many accounts say that there was tension between Leibniz's work and Newton's work, and in particular there was tension between conservation of vis-viva and Newtonian mechanics. (Of course, we now know there is no tension between the two things.)

However, I couldn't find any specific comments by Isaac Newton on the concept of Leibniz's vis-viva. Are there any quotes by Isaac Newton himself where he explicitly rejects this notion as meaningful? Does he ever make any reference to it?

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The controversy was not so much about the tension between vis viva and mechanics, as about what is the "true" quantity of motion, vis viva or momentum, and what is the "metaphysical" basis upon which mechanics is to be built, vis viva or Newton's force (aside from multiple other side issues that got entangled with it, see What was the vis viva controversy, including its philosophical aspects?). And, as is often the case, Newton's contribution was not by what he said, but by what he implied in multiple editions of Principia, see Smith, The vis viva dispute. For example, he highlighted the non-conservation of vis viva in soft collisions, and tied its conservation to static ("dead") forces, like the centripetal force, in direct counter to Leibniz's use of it to distinguish "dead" and "living" forces:

"Book 1 of Newton’s Principia, the part that has some bearing on the vis viva controversy, went to the printer in April 1686, too soon for him to have seen the Acta Eruditorum issue containing Leibniz’s note. Newton surely was aware of the controversy by the time of the second (1713) and third (1726) editions of the Principia; yet they never mention it. Nevertheless, parts of Book 1 that remained the same in all editions did feed the controversy. For example, the conservation of momentum is presented as a corollary of Newton’s laws of motion, with Huygens’s center-of-gravity principle, carefully defended, as the next corollary.

In his empirical defense of his laws of motion, Newton indicates how to make corrections for air resistance in measurements of a ballistic pendulum (see figure 2(a)) to obtain more exacting tests of theories of collision, and then adds that similar corrections can be made for imperfect elasticity of the colliding bodies. Thereby Newton underscores the failure of mass times velocity squared to be conserved when the bodies are not perfectly hard.

[...] Leibniz’s 1686 note provoked exchanges with the Cartesians. Descartes’ conservation of motion (see figure 1) was difficult to abandon if one believed that all space is filled with matter. The exchanges led Leibniz to refine his position in writings on “dynamics” (the term is his) that were not published until the 19th century. In those writings Leibniz grants the conservation of directional motion, but argues that because it is directional, unlike $mv^2$, it involves reference to other bodies and therefore is not a feature of each body taken unto itself. He concedes that $mv^2$ is not obviously conserved in the collision of soft bodies. But he contends that it is actually conserved via undetected motion of the microphysical parts of the bodies...

Invoking the metaphysical principle that the effect must equal the cause, Leibniz gave a variant of his 1686 argument: He calculated “the force through the effect produced in using itself up” to conclude that the force transferred from one equal body to another varies as the square of the velocity. Leibniz made clear that the metaphysical principle is what establishes the priority of the conservation of living forces in changes of motion."

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  • $\begingroup$ This is very informative! I've been reading various things about this. To be absolutely clear, is it true that Newton was aware of Hyugen's, Mariotte's, and Wrenn's work on both elastic and inelastic collisions? In other words, Leibniz and Newton had no disagreements what happens with hard collisions. Is that correct? $\endgroup$ Jan 16 at 5:30
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    $\begingroup$ @MaximalIdeal He sure was. Wren's paper was published in the Transactions of the Royal Society, and Huygens's in Journal des Sçavans back in 1669, Principia builds on the results, and is modeled on Huygens's exposition style generally. They had no disagreements on what is observed in either hard or soft collisions, but on what "metaphysically" happens in mechanics generally, and in collisions in particular, disagreements were aplenty. For Leibniz, vis viva gets passed around, for Newton, forces act and react. $\endgroup$
    – Conifold
    Jan 16 at 5:48

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