In symbols, U, as introduced by German physicist Rudolf Clausius in 1850, represents the energy or internal energy of a body or system. [1]

The symbol U, prior to its adoption by Clausius in thermodynamics, has deep roots, which, of course, guided Clausius to this symbol use. Specifically, Clausius utilized Irish mathematician William Hamilton’s 1835 assignment, from his General Method in Dynamics, of U as the ‘force function’ of a system (which Clausius called the ‘ergal’), loosely defined as the sum of the forces of attractions and repulsions acting on the material points of any system (i.e. work in a modern sense), but added to it German mathematician Gottfried Leibniz’s 1686 vis viva, so to re-define the energy U of a body or system as the sum of the vis viva T plus the ergal J: [2]

U = T + J

Prior to this, subsequently, Clausius’ internal energy, has its roots in both Leibniz’s vis viva and Hamilton’s force function, the latter of which has its roots in Joseph Lagrange’s 1788 ‘central function’, from his Analytical Mechanics treatise. [3]

1. Clausius, Rudolf. (1879). The Mechanical Theory of Heat (etymology of U: pg. 31). London: Macmillan & Co.
2. (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.
3. (a) Lagrange, Joseph. (1853). Mecanique Analytique (Vol. 1). Publisher.
(b) Lagrange, Joseph. (1855). Mecanique Analytique (Vol. 2). Publisher.

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