What is the geometric meaning of Cauchy's functional equation?
$$f(x+y) = f(x)+f(y) \quad \forall x,y$$
Some of the early work on this equation by Darboux and Hamel may have been partially motivated by considerations from physics, but I've never really looked into it and virtually all of the vast literature on Cauchy's functional equation seems to ignore this possible connection. The papers I know about are listed below in case you or someone else wants to look into this.
Jean Gaston Darboux, Sur la composition des forces en statique [On the composition of forces in statics], Bulletin des Sciences Mathématiques et Astronomiques (1) 9 (1875), 281-288.
Jean Gaston Darboux, Sur le théorème fondamental de la géométrie projective. (Extrait d'une lettre à M. Klein) [On the fundamental theorem of projective geometry. (Excerpt of a letter to Mr. Klein)], Mathematische Annalen 17 #1 (1880), 55-61.
Georg Karl Wilhelm Hamel, Über die zusammensetzung von vektoren [On the composition of vectors], Zeitschrift für Mathematik und Physik 49 (1903), 362-371.
(ADDED 3 MONTHS LATER) Entirely by accident, this morning I happened to come across a paper in English that I strongly suspect would be useful in connection with the papers above. Although I have not had a chance to look over it yet, and might not for some time, I thought it would be useful to mention it.