forcIn thermodynamics, force function, or force-function, is the negative of the potential energy of a given system or body.

History
In 1788, Italian mathematician Joseph Lagrange introduced a ‘central function’ in his Analytical Mechanics. [4]

In 1835, supposedly building on Lagrange, Irish mathematician William Hamilton, in his General Method in Dynamics, introduced what he called a "force function" of a system, a function U of the masses and mutual distances of the material points of a given system. [3]

In circa 1841, German mathematician Carl Jacobi introduced a “force function”, which according to the retrospect 1882 view of Hermann Helmholtz, is said to be equal to the negative of the potential energy (a William Rankine term, 1853) of the system. [1]

In the work of German physicist Rudolf Clausius (1850-1865), Hamilton's version of the force function was cited in the formulation of the thermodynamic concept of internal energy, symbol U. [6]

In 1876, American engineer Willard Gibbs was defining giving the following two definitions: [2]

Characteristic function notation
Modern notation
Description
Name assigned by Gibbs (modern shorthand name)
$- \epsilon \,$ $- U \,$ Negative of the internal energy of a system ‘force-function for constant entropy’ (isentropic force function)
$- \psi \,$ $- H \,$ negative of the Helmholtz free energy of the system ‘force function for constant temperature’ (isothermal force function)

Characteristic function notation	Modern notation	Description	Name assigned by Gibbs (modern shorthand name)
$- \epsilon \,$	$- U \,$	Negative of the internal energy of a system	‘force-function for constant entropy’ (isentropic force function)
$- \psi \,$	$- H \,$	negative of the Helmholtz free energy of the system	‘force function for constant temperature’ (isothermal force function)

In 1882, German physicist Hermann Helmholtz stated that his “quantity of tension force” was essentially the same as Jacobi’s force function as well as Rudolf Clausius’ ergal. [1]

In 1923, American physical chemist Gilbert Lewis stated: [5]

“We may think of the quantity ‘– ΔG ’ as the driving force of a reaction; where, in a thermodynamic sense, a system is stable when no process can occur with a diminution in free energy.”

Which seems to be in alignment with Helmholtz 1882 article section "Idea of Free Energy", wherein he states that the force function is the negative of the potential energy, in the sense that Gibbs free energy is often defined as the isothermal-isobaric thermodynamic potential and also the driving force for earth-bound natural processes.

References
1. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes”, in: Physical Memoirs Selected and Translated from Foreign Sources, 1: 43-97. Physical Society of London, Taylor and Francis, 1888.
2. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances" (force-function, pgs. 89, 204, 335), Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.
3. (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.
4. (a) Lagrange, Joseph. (1853). Mecanique Analytique (Vol. 1). Paris.
(b) Lagrange, Joseph. (1855). Mecanique Analytique (Vol. 2).
5. Lewis, Gilbert N. and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances (pgs. 160-61). McGraw-Hill Book Co., Inc.
6. Clausius, Rudolf. (1879). The Mechanical Theory of Heat (etymology of U: pg. 31). London: Macmillan & Co.