I am not sure if I would call the following a definitive answer. Rather it's a hypothesis with evidence. My starting point was to explore the folklore idea (example 1, 2) that the root of "p" is from German physics ("impuls" which indeed is a derivative of pulsus for momentum. That the notion of impuls for momentum has its own problematic history might be a hint that this folklore idea might not hold.)
Up until the 9th edition, one of the most popular German physics textbooks now known as Gerthsen's Physik used $J$ which was published in 1966:
In the 10th edition of 1969 this was changed to $p$ as follows (also in this edition all forms of German Fraktur were removed):
That this is not an ideosyncracy of Gerthsen can be seen in other textbooks such as Westphal of 1956:
It is rather common to encounter momentum defined equationally but without symbol, i.e. as $mv$ often in context where $p$ clearly stands for something else. From Mach, E. (1889) Die Mechanik in ihrer Entwicklung, p.344:
$p$ here is pressure and $mv$ is momentum. An example where $p$ is force and $mv$ has no letter assigned can also be found in E. Mach (1919) The Science of Mechanics. Transl. McCormack, Open Court. p. 249.
I submit that all these examples make the idea that German physics being the root of $p$ for momentum rather implausible.
A first hint at my hypothesis can be found in:
- Sommerfeld, A. (1942) Vorlesungen ueber Theoretische Physik, Band 1 - Mechanik. Springer.
On p.113, one finds discussion the Fraktur symbol 𝕲 for total momentum as follows:
However, later in the same source on page 177:
"Lagrangesche Impulskoordinaten" translates to "Lagrangian momentum coordinates". Hence Sommerfeld says that letters $p_1,\ldots,p_n$ are to be reserved for that purpose.
This suggests Lagrange as the source. Indeed this is the core of my hypothesis.
Lagrange however, did use the variable $q$ for what we now call generalized coordinates. This had no special meaning, rather Lagrange liked to use certain sequences of letters in his exposition, such as $X,Y,Z$, $P,Q,R$, $l,m,n$, $a,b,c$ and Greek letter sequences. The choice of $Q$ for the typical "generalized coordinates" appears quite arbitrary.
Hamilton writes the following in his
- Hamilton, W. R. (1834). On a general method in dynamics. Phil. Trans. Royal Soc., pp. 247–308.
The only mention of momentum is contained in this context (p. 257):
Hence Hamilton does not assign the letter $p$ to momentum, instead spells out expressions essentially of the form $mv$.
The first explicit link of $p$ to some notion of impulse or momentum - known to me - is Gibbs in:
- Gibbs, J. W. (1879). On the fundamental formulae of dynamics. American Journal of Mathematics, 2(1), 49-64.
when on page 60 he writes:
Though certainly the use of letters $p$ and $q$ have been transported through other authors such as Jacobi (see his 1884 Vorlesungen ueber Dynamik).
To summarize, my hypothesis is that the letter $p$ and $q$ are original to Lagrange but not with $p$ explicitly as momentum. The letter use persisted and was later (possibly Gibbs), explicitly linked to momentum, and without the letter reference conceptually linked to momentum by Hamilton.