I am unaware if someone has asked this before, but I am studying classical mechanics and I don’t know why do we use $U$ for potential energy. I have read that Rankine used it first, but I can’t find any explanation for this.


It is possible that $U$ stands for "utility" (in mechanics), and it is due to Hamilton and not to Rankine. Possibly, Rankine wished to be consistent with Hamilton's notation, but, in any case, I suspect hat the modern use in mechanics has more to do with following Hamilton than with following Rankine, whose musings are very abstract and metaphysical, and are not even directed at mechanics. He does not even mention kinetic energy, for example.

Rankine's paper On the general law of the transformation of energy (1853) that introduces potential energy explicitly gives no explanation, "Let $U$ denote this potential energy" is all he wrote. It should be noted that the concept of potential energy is much older. It appears, in a rudimentary form, in Leibniz's writings under the name of "motive force", along with a prototype of mechanical conservation law, which Rankine spelled out explicitly two centuries later, see Leibniz and the Vis Viva Controversy by Iltis.

Lagrange used $V$ for potential energy in Mécanique analytique (1788-9), possibly from Latin vis, and $T$ for kinetic energy. Hamilton kept Lagrange's $T$, but, for whatever reason, switched from $V$ to $U$ in On a General Method in Dynamics (1834), which introduces what we now call Hamiltonian dynamics. Of course, neither of them called it "potential energy", but the letters occupy the familiar places in deriving equations of motion from a variational principle. Hamilton introduces

"A function $U$ of the masses and mutual distances of the several points of the system, of which the form depends on the laws of their mutual actions, by the equation $U=\sum. mm_{'}f(r)$, $r$ being the distance between any two points $m,m_{'}$, and the function $f(r)$ being such that the derivative or differential coefficient $f'(r)$ expresses the law of their repulsion, being negative in the case of attraction. The function which has been here called $U$ may be named the force-function of a system: it is of great utility in theoretical mechanics, into which it was introduced by Lagrange, and it furnishes the following elegant forms for the differential equations of motion... the second members of these equations being the partial differential coefficients of the first order of the function $U$".

The only word in this passage that suggests a reason for denoting the "force-function" $U$ is "utility".

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