Interactions between mechanics and differential geometry in the 19th century

Abstract

79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19th century ideas discussed here influenced the developments in the 20th century. Einstein himself is a hotly debated case. Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20th century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz. Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions. © 1995 Springer-Verlag.