In science, vis viva, defined in Latin as “living force”, as contrasted with vis mortua, i.e. "dead force", is the mathematical quantity of the product of the mass of a moving object multiplied by its velocity squared, mv². [1] This quantity was determined to be conserved absolutely in perfectly elastic collisions, such as between steel balls, by Dutch physicist Christiaan Huygens in 1669. [4] This was a precursor to kinetic energy.

History
See main: Vis viva dispute
In circa 1640, French mathematician René Descartes ruminated on the issue of what happens when a body, such as a hard spherical ball, of mass m and speed v collides with another such body of different mass and speed. He concluded that the quantity mv (now called momentum) would be conserved. In other words, according to Descartes, for two colliding spheres the sum:

$m_1 v_1 + m_2 v_2 \,$

would be the same before and after the collision. In short, Descartes argued for a principle of conservation of motion, where quantity of the product of the mass of a moving object by its velocity mv was thought to be conserved in mechanical interactions. Descartes’ reasoning, however, was partly metaphysical and therefore not convincing. [5]

In the 1660s, Dutch physicist Christiaan Huygens, an associate of Descartes, carried out many ingenious experiments on colliding spheres, some of which were demonstrated at the Royal Society in London. This led him, in 1669, to the correct conclusion that the quantity preserved in a collision is not mv but mv².

In 1686, in opposition to Descartes' logic, German mathematician Gottfried Leibniz, coined the name "vis viva" for Huygens' quantity
mv², and argued for a principle of the conservation of vis viva. [2]

½ factor
In 1811, according to one reference, Italian mathematician Joseph Lagrange used calculus to show that a factor of two is involved in the relationship “potential” (potential energy) and “vis viva” (kinetic energy). [2] Conversely, according to a second reference, it was French physicist Gustave Coriolis who in 1829 introduced the factor ½ in Leibniz’s vis viva for the sake of mathematical convenience. [7]

Vis viva = $\tfrac12 m v^2 \,\!$

Vis viva + vis mortua
In circa 1740, French physicist Emilie du Chatelet suggested that living force or vis viva could be converted into vis mortua or dead essence and that the sum of the two quantities, being interconvertable, would remain constant. [6] Vis mortua is said to have been a precursor to potential energy, just as vis viva was to kinetic energy. [4]

These two entities, vis viva and vis mortua were taken up later by Italian mathematician Joseph Lagrange, in his 1788 Analytical Mechanics, who enunciated, in effect, that the total mechanical energy, composed of kinetic and potential energies, is constant in an isolated system. This work was carried on further by Irish mathematician William Hamilton in his 1834 On a General Method in Dynamics; work later cited by German physicist Rudolf Clausius in his development of internal energy.

Kinetic energy
In 1847, English physicist James Joule highlighted the name association issues: [8]

"The force expended in setting a body in motion is carried by the body itself, and exists with it and in it, throughout the whole course of its motion. This force possessed by moving bodies is termed by mechanical philosophers vis viva, or living force. The term may be deemed by some inappropriate, inasmuch as there is no life, properly speaking, in question; but it is useful, in order to distinguish the moving force from that which is stationary in its character, as the force of gravity. When, therefore, in the subsequent parts of this lecture I employ the term living force, you will understand that I simply mean the force of bodies in motion.

The term vis viva, however, was soon abandoned in favor of first, for a few tentative years, the term "actual energy" by William Rankine, in 1853, and then eventually "kinetic energy" by William Thomson and Peter Tait in 1862 [9]

Thermodynamics
In thermodynamics, vis viva was a model of logic in precursory form to the principle of the conservation of energy; used in theory development by those such as Scottish engineer William Rankine and German physicist Rudolf Clausius. By 1879, Clausius had come to define the vis viva, symbol T, for a whole system of points as: [3]

$T = \sum \frac {m} {2} v^2$

References
1. Hesse, Mary B. (1962). Forces and Fields – the Concept of Action at a Distance, (pgs. 159, 161, 163). New York: Dover.
2. George E. Smith. (2006). "The Vis Viva Dispute: A Controversy at the Dawn of Dynamics", Physics Today 59, Oct., Issue 10, pp 31-36.
3. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (pg. 18). London: Macmillan & Co.
4. Hokikian, Jack. (2002). The Science of Disorder: Understanding the Complexity, Uncertainty, and Pollution in Our World (pg. 3-4). Los Feliz Publishing.
5. Laidler, Keith J. (2002). Energy and the Unexpected, (pg. 22). Oxford University Press.
6. Émilie du Châtelet – Wikipedia.
7. (a) Coriolis, Gustave. (1829). Calculation on the Effect of Machines, or Considerations on the use of Motors and their Evaluation (Calcul de l'Effet des Machines, Ou Considerations sur l’emploi des Moteurs et sur Leur Evaluation). Paris.
(b) Jammer, Max. (1957). Concepts of Force: a Study in the Foundations of Dynamics (pgs. 166-67). Harvard University Press.
8. Joule, James. (1847). “On Matter, Living Force, and Heat”, Lecture at St. Ann’s Church Reading room; in: Manchester Courier newspaper, May 5 and 12; in The Scientific Papers, Volume 1 (pg. 266). The Physical Society, Great Britain.
9. Maxwell, James. (1871). Theory of Heat (pg. 91). Longmans.