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Rudolf Lipschitz’s work on differential geometry and mechanics. (English) Zbl 0809.01010
Knobloch, Eberhard (ed.) et al., The history of modern mathematics. Volume III: Images, ideas, and communities. Boston, MA: Academic Press. 113-138 (1994).
In the development of mechanics, the study of differential geometry and the new concepts of manifold and curvature introduced by Riemann in 1854 (in his Habilitationsvortrag) represented a fruitful starting point; in this manner, the principles of mechanics have been extended to a Riemannian manifold provided with a nonzero curvature tensor. In this respect, mention should be made of Lipschitz, Schering and Killing, who formulated a potential theory in an $$n$$-dimensional space with constant curvature, thus evidencing the profound relationship between mechanics and geometry. Riemannian manifolds have been treated in two articles, published in Crelle’s Journal, written by Lipschitz and Christoffel, trying to solve one of the questions left open by Riemann; yet, while Christoffel was interested only by the analytical aspects involved Lipschitz was looking for the mechanical meaning by linking the solution to the integration of a system of differential equations of motion. Lipschitz extended the Riemann curvature tensor to handle quadratic forms with $$p$$ greater than 2. Also, he developed the mechanical implications of his studies with some of Hamilton’s ideas and solutions; he extended the Hamiltonian principal function to a manifold whose metric tensor was a homogeneous positive-definite form. As to the relations existing between geometry and mechanics, Lipschitz considered that not only some of the geometrical statements enunciated by Gauss, Riemann and Beltrami could be considered as mechanical laws, but they even were, actually, the very mechanical concepts permitting a profound analysis of the corresponding geometrical relations. Another domain of interest for Lipschitz was the application of the general results on the integration of the Euler-Lagrange differential equations to the problem of attraction, according to the Newtonian law, or the planet-problem.
For the entire collection see [Zbl 0799.00010].
Reviewer: C. Cusmir (Iaşi)

##### MSC:
 01A55 History of mathematics in the 19th century