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Leibniz's concept of vis visa (literally translated as living force) was a precursor to our modern concept of kinetic energy. His formula for it was close to the modern non-relativistic one: $mv^2$, but not including factor $\tfrac{1}{2}$.

Vis viva became a bone of contention between the Newtonians and the Cartesians in the 18th century; historians call this the vis viva debates. The debate mixed technical, philosophical, and even theological issues in a manner that seems very strange to modern sensibilities. However, this kind of argument was characteristic of the time.

Can someone provide a summary?

Note: the modern concept of energy (including conservation) only emerged fully in the 19th century, thanks to the work of Helmholtz, Meyer, Joule, William Thomson, Clausius, and others. However, that is a topic for another question (maybe this one).

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    $\begingroup$ You have to start from Descartes' Physics; see Ch.4 The Laws of Motion and the Cartesian Conservation Principle. Descartes formulated in his Principles (1644) three laws of of bodily motion, the third one regarding the collisions of bodies. According to D, "the quantity conserved in collisions equals the combined sum of the products of size and speed of each impacting body." The use of the quantity of motion as magnitude to be conserved, lead D to some paradoxcial results : 1/2 $\endgroup$ – Mauro ALLEGRANZA Nov 30 '14 at 18:07
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    $\begingroup$ "Descartes claims that a smaller body, regardless of its speed, can never move a larger stationary body." See Leibniz's Philosophy of Physics for Leibniz's refutation of Cartesian Laws of Motion : "Leibniz is therefore able to argue that Descartes's laws of motion are untenable because they would lead to violations of the conservation of force as measured by $mv^2$ [a quantity that L called vis viva]." 2/2 $\endgroup$ – Mauro ALLEGRANZA Nov 30 '14 at 18:13
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    $\begingroup$ @MauroALLEGRANZA Richard J. Blackwell's article, "Descartes' Laws of Motion" (Isis, Vol. 57, No. 2 (Summer, 1966), pp. 220-234) is a nice treatment of this. $\endgroup$ – Michael Weiss Nov 30 '14 at 18:17
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    $\begingroup$ See also Émilie du Châtelet for the first successful "merge" of the Newtonian tradition (momentum) with the Leibnizian one. $\endgroup$ – Mauro ALLEGRANZA Nov 30 '14 at 18:19
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    $\begingroup$ I sugegst you Daniel Garber's studies : his contribution on D's physics in The Cambridge Companion to Descartes and his books : Descartes' Metaphysical Physics, 1/2 $\endgroup$ – Mauro ALLEGRANZA Nov 30 '14 at 18:22
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The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first, a "skillful attack by Leibniz against an inadequate concept, $m|v|$, and its description of the world", it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony.

After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is Mach's summary: "Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum $\sum mv$ is a constant, and the investigations of Hyugens showed that also the sum $\sum mv^2$ is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743". Some modern scholars question this conclusion however, pointing out that the controversy lingered on "through the remainder of the eighteenth century", that a crucial observation on the issue only appears in the second edition of Traite de Dynamique (1758), and was made earlier by Boscovich (1745). Namely, "vis viva is the measure of a force acting through a distance while momentum is the measure of a force acting through a time".

The philosophical side was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the "source of the vitality" of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz.

The starting point of the controversy was Descartes defining "momentum" as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Cartesian "momentum" is not conserved. By that time Huygens already established that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of "Newton's" second law), so Leibniz declared that the "true quantity of motion" and called it vis viva. In the meantime Wallis gave the correct description of what happens to velocities in elastic collisions, which is equivalent to "conservation of momentum" (he uses no such language, and there was no notion of vector at the time to define "momentum"). Newton incorporated that into Principia but continued to call Cartesian "momentum", you guessed it, "quantity of motion". And so it began.

Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, separate issues got entangled with the original controversy. In 1724 Paris Academy offered a prize for the "best" way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling mv the "force of bodies", and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are "better" for mechanics, Newton's or Leibniz's.

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I must put up a clarifying answer. They groped towards different concepts and talked past each other, in the modern perspective.

Much of what we call Newtonian mechanics is due to Euler, who called it such in his early papers on the topic, but Euler wrote after the controversy. (See Truesdell)

The problem is, as is clearly seen from the primary texts of Newton and Leibnitz, that Newton was working with the concept we call force (indeed, in the later edition of his optics, he discovered repulsive forces, in addition to attractive forces he postulated). This would lead to the progress made by Euler and Boscovich. Cartesius worked with momentum prior both.

Leibnitz was however concerned primarily with mechanics only so far as it concerned his "the most diverse one the one possible world that exists" philosophy. He argued that since all laws of nature can be written (truly modelled analogously) as equalities, the "whole effect" is always conserved. One side is cause and the other effect. This was intended to eliminate the concept and problem of "first cause" from an practical significance in his philosophical system of the world. Which side in an equation is the whole cause and which is the whole effect?

This is loosely similar to the modern position, because we no longer care or even assume existence of first causes, and it is not impossible that a time reverse situation occurs—for the cause and the effect in this case are in an equivalence class—but rather it is highly improbable, as determined by the thermodynamics of the situation. From here we have Boltzmann and Mach picking up the stick.

(The confusion early on was caused by the fact that modern theory in thermodynamics did not exist prior to Waterston, Mayer, and Boltzmann. Indeed, Leibnitz could not explain where the "effect" went when a body dropped from a height fell onto the floor, which ended its fall. That was the Mayer experiment 150 years later. And 15 years after than, even scientists such as Kelvin still had to be shown errors in their papers where they claimed they'd found violations of energy conservation—such as when Waterson pointed out to Kelvin that phase transformations do in fact require energy and cannot be ignored in well constructed experiments. The energy conservation concept was broadly spread at that point by the popular Spencer with his synthetic philosophy.)

Leibnitz simply extrapolated from a pendulum experiment that $mv^2$ is conserved in all cases. Such extrapolation is not logically kosher, true, but that was his hypothesis in his dynamics. He was essentially groping toward the modern energy concept, which was finally clarified by Mayer and Clausius and Mach and Einstein, in that order. It was Mayer who gave the first clear discussion of kinetic as distinguished from potential as distinguished from thermal energy. Maupertuis and D'Alembert and Euler worked on clarifying the concepts of kinetic and potential energy.

Without thermodynamics the controversy could not have been resolved given the state of knowledge when it occurred—and so indeed it wasn't. There are no miracles.

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