Interactions between mechanics and differential geometry in the \(19^{th}\) century
According to the author, this is not an exhaustive study (p. 64). There are established two main periods of interaction, one before 1870 and one after 1870. Before 1870 there only existed some scattered interactions. The starting point is the year 1828. Following the Lagrangian tradition Hamilton tried to conquer optics by means of an analytical formalism; in the same year Gauss published his intrinsic differential geometry. First steps into the main direction were made by Cayley who made use of 3-dimensional projective geometry in the kinematics of rigid bodies (1843), and Schläfli who introduced \(n\)-dimensional geometry in general dynamics (1852/3). A main contribution is due to Jacobi in 1837 who removed the Hamilton formalism even further from geometry than Hamilton and initiated the geometrisation of the principle of least action. Liouville followed this idea and, in using generalized coordinates, he came close to a geometrisation of the \(n\)-dimensional configuration space and a generalisation of the principle of least action. Lipschitz opened a new era, his goal was a description of dynamics on the basis of differential geometry, i.e. differential forms. Now, configuration space became a Riemannian space in \(n\) dimensions. Darboux was successful in giving these ideas a clearer presentation so that now mechanics really were an integral part of \(n\)-dimensional Riemannian geometry. On the other hand Paul Stäckel contributed in studying transformations of mechanical systems. The new geometry caused consequences in physics. As examples some of the ideas of Zöllner, Clifford and Hertz are discussed. The main result is that the interaction of mechanics and differential geometry is not a 20th century invention. Similar results were achieved earlier by R. Ziegler in 1985 and R. Tazzioli in 1993/4. It is quite interesting because Ziegler and Tazzioli were not investigating exactly the same developments. So the present paper is complementing the previous results.
scientific article; zbMATH DE number 794324
scientific article