Vis viva (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy.

## Overview

Proposed by Gottfried Leibniz over the period 1676–1689, the theory was controversial as it seemed to oppose the theory of conservation of quantity of motion advocated by René Descartes.[1] Quantity of motion is different from momentum. However, Newton defined quantity of motion as the conjunction of the quantity of matter and velocity (see Definition II in Principia). In Definition III he defines the force which resists a change in motion as the vis inertia of Descartes. His Third Law of Motion is a statement of what becomes known as the conservation of momentum as he demonstrates in the related Scholium. Leibniz accepted the principle of conservation of momentum, but rejected the Cartesian version of it.[2] The difference between Newton and Descartes and Leibniz was whether the quantity of motion was simply related to a body's resistance to a change in velocity (vis inertia) or whether a body's amount of force due to its motion (vis viva) was related to the square of its velocity.

The theory was eventually absorbed into the modern theory of energy though the term still survives in[3] the context of celestial mechanics through the vis viva equation. The term "living force" was also used, for example by George William Hill.

The term is due to German Gottfried Wilhelm Leibniz, who during 1676–1689 first attempted a mathematical formulation. Leibniz noticed that in many mechanical systems (of several masses, mi each with velocity vi) the quantity:[4]

${\displaystyle \sum _{i}m_{i}v_{i}^{2))$

was conserved. He called this quantity the vis viva or "living force" of the system.[4] The principle, it is now realised, represents an accurate statement of the conservation of kinetic energy in elastic collisions, and is independent of the conservation of momentum.

However, many physicists at the time were unaware of this fact and, instead, were influenced by the prestige of Sir Isaac Newton in England and of René Descartes in France, both of whom advanced the conservation of momentum as a guiding principle. Thus the momentum:[4]

${\displaystyle \,\!\sum _{i}m_{i}\mathbf {v} _{i))$

was held by the rival camp to be the conserved vis viva. It was largely engineers such as John Smeaton, Peter Ewart, Karl Holtzmann, Gustave-Adolphe Hirn and Marc Seguin who objected that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston.

The French mathematician Émilie du Châtelet, who had a sound grasp of Newtonian mechanics, developed Leibniz's concept and, combining it with the observations of Willem 's Gravesande, showed that vis viva was dependent on the square of the velocities.[5]

Members of the academic establishment such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics but in the 18th and 19th centuries, the fate of the lost energy was still unknown. Gradually it came to be suspected that the heat inevitably generated by motion was another form of vis viva. In 1783, Antoine Lavoisier and Pierre-Simon Laplace reviewed the two competing theories of vis viva and caloric theory.[1] Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat. Vis viva now started to be known as energy, after the term was first used in that sense by Thomas Young in 1807.

An excerpt from Daniel Bernoulli's article, published in 1741,[6] with the definition of vis viva with 12 multiplier.

The recalibration of vis viva to include the coefficient of a half, namely:

${\displaystyle E={\frac {1}{2))\sum _{i}m_{i}v_{i}^{2))$

was largely the result of the work of Gaspard-Gustave Coriolis and Jean-Victor Poncelet over the period 1819–1839, although the present-day definition can occasionally be found earlier (e.g., in Daniel Bernoulli's texts).

The former called it the quantité de travail (quantity of work) and the latter, travail mécanique (mechanical work) and both championed its use in engineering calculation.

## Notes

1. ^ McDonough, Jeffrey K. (2021), "Leibniz's Philosophy of Physics", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.), Metaphysics Research Lab, Stanford University, retrieved 2021-08-28
2. ^ McDonough, Jeffrey K. (2021), "Leibniz's Philosophy of Physics", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.), Metaphysics Research Lab, Stanford University, retrieved 2021-08-28
3. ^
4. ^ a b c Smith, George E. (October 2006). "The vis viva dispute: A controversy at the dawn of dynamics". Physics Today. 59 (10): 31–36. doi:10.1063/1.2387086. ISSN 0031-9228.
5. ^ Musielak, Dora (2014). "The Marquise du Chatelet: A Controversial Woman of Science". arXiv:1406.7401 [math.HO].
6. ^ Bernoulli D. (1741). "De legibus quibusdam mechanicis...". Commentarii Academiae Scientiarum Imperialis Petropolitanae. 8: 99–127.