In science, Thomas Young (1773-1829) was an English physician-physicist notable for being the first to use and define the term “energy” in the modern physical sense. [1]

Overview
The modern physics meaning of the term is generally attributed to English physicist and physician Thomas Young who in 1807 used the term energy, based on the Greek word ένεργεια meaning efficacy or effective force, as an abbreviation for the sum of kinetic energy and gravitational potential energy of a mass and the elastic energy of a spring to which the mass may be attached. [2] In more detail, under the guidance of American-born English physicst Benjamin Thomson, in 1802 Young began to prepare lectures in natural philosophy for the Royal Society. In them, he dealt with the quantity “vis viva” and in 1807 introduced the term energy for it: [3]

“The term energy may be applied, with great propriety, to the product of mass or weight of a body, into the square of the number expressing its velocity. Thus, if the weight of one ounce moves with a velocity of a foot in a second, we call its energy 1; if a second body of two ounces has a velocity of three feet in a second, its energy will be twice the square of three, or 18.”

In this sense, Young defined energy E as:

E = mv²

To clarify that Young was the first to have mathematically coined the term "energy", years later Scottish mathematical physicist William Thomson stated before an audience, for instance, that “the very name energy, though first used in its present sense by Thomas Young about the beginning of this century, has only come into use practically after the doctrine which defines it had ... been raised from mere formula of mathematical dynamics to the position it now holds of a principle pervading all nature and guiding the investigator in the field of science.” [4]

Kinetic energy
In 1811, in partial correction to Young’s expression, 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). [3] In the symbols used by Lagrange, i.e. T as kinetic energy, in his 1788 Analytical Mechanics:

where 2T denotes the whole "living force of the system". [5]

References
1. Fechner, Gustav Theodor (1878). Ueber den Ausgangswerth der kleinsten Absweichungssumme. S. Hirzel.
2. Muller, Ingo. (2007). A History of Thermodynamics - the Doctrine of Energy and Entropy. New York: Springer.
3. Rayner, John, N. (2000). Dynamic Climatology: Basis in Mathematical Physics, (pg. 94). Blackwell Publishing.
4. Thomson, William. (1881). "On the Sources of Energy Available to Man for the Production of Mechanical Effect." BAAS Rep. 51: 513-18 (Quote: pg. 513); PL 2: 433-50.
5. (a) Hamilton, W.R. (1834). “On a general method in dynamics by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.” Philos. Trans. R. Soc. London, 124:247-308.
(b) Hamilton, W.R. (1835). “A second essay on a general method in dynamics.” Philos. Trans. R. Soc. London, 125:95-144.

External links
● Thomas Young – Wikipedia.