Interactions between mechanics and differential geometry in the \(19^{th}\) century
type: Item
instance of: scholarly article
title: Interactions between mechanics and differential geometry in the \(19^{th}\) century
zbMATH Open document ID: 0834.01007
DOI: 10.1007/BF00374699
published in: Archive for History of Exact Sciences
publication date: 1995-10-18T00:00:00Z
review text: 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.
reviewed by: Karin Reich
Mathematics Subject Classification ID: 01A55
zbMATH DE Number: 794324
zbMATH Keywords: principle of least action
zbMATH Keywords: geodesics
zbMATH Keywords: non-Euclidean dynamics
zbMATH Keywords: trajectories
zbMATH Keywords: \(n\)-dimensional geometry
zbMATH Keywords: Lipschitz
zbMATH Keywords: optics
Wikidata QID: Q56506420
author: Jesper Lützen
MaRDI profile type: Publication
cites work: The mathematical technique in Fermat's deduction of the law of refraction
cites work: Poincaré's discovery of homoclinic points
cites work: Q4167627
cites work: Oscar II's prize competition and the error in Poincaré's memoir on the three body problem
cites work: XXIX. On the analytical theory of the conic
cites work: Q5800805
cites work: The missing link: Riemann's ``Commentatio'', differential geometry and tensor analysis
cites work: Q4478108
cites work: Q3874881
cites work: C. G. J. Jacobis Vermächtnis einer ‘konventionalen’ analytischen Mechanik: Vorgeschichte, Nachschriften und Inhalt seiner letzten Mechanik-Vorlesung
cites work: Die Geschichte der Differentialgeometrie von Gauß\ bis Riemann (1828--1868)
cites work: Q4040685
cites work: Q3879193
cites work: Ether and theory of elasticity in Beltrami's work
cites work: Q4312140
cites work: On Helmholtz' paper ``Über die thatsächlichen Grundlagen der Geometrie''
cites work: Q3701378
full work available at URL: https://doi.org/10.1007/bf00374699
OpenAlex ID: W2318003355
description: scientific article; zbMATH DE number 794324