Abstract
The modern geometric viewpoint in Physics owes a great deal to Lagrange’s Analytical Mechanics (Lagrange 1788). In Lagrange’s view, a mechanical system is characterized by a finite number n of degrees of freedom to each of which a generalized coordinate is assigned. A configuration of the system is thus identified with an ordered n-tuple of real numbers. What are these numbers coordinates of? An answer to this question, which could not have been, and was in fact not, asked at the time, would have brought Lagrange close to the general concept of topological manifold.
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- 1.
Riemann’s work is translated and reproduced in its entirety in Spivak (1979), itself an invaluable source for the study of differential geometry.
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This notion will be made more precise when we define a differentiable manifold and its tangent bundle.
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For the use of groupoids in the theory of material uniformity see Epstein and de León (1998).
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Epstein, M. (2014). Physical Illustrations. In: Differential Geometry. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-06920-3_2
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DOI: https://doi.org/10.1007/978-3-319-06920-3_2
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