Airysche Spannungsfunktion und isotrope Differentialgeometrie
type: Item
instance of: scholarly article
title: Airysche Spannungsfunktion und isotrope Differentialgeometrie
zbMATH Open document ID: 0106.35802
DOI: 10.1007/BF01195167
author: Karl Strubecker
published in: Mathematische Zeitschrift
publication date: 1962-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/170022
review text: In the classical plane elasticity theory the Airy function \(z(x, y)\) plays an important role satisfying the equation \(\Delta\Delta z(x, y) = 0\), The author observes that this equation is invariant with respect to the group \(G_6\) of transformations \[x' = a + x \cos \varphi - y \sin \varphi,\quad y' =b + x \sin \varphi + y \cos \varphi,\quad z' = c + c_1x + c_2y +z \] which defines the isotropic geometry already developed by the same author [Math. Z. 47, 743--777 (1942; Zbl 0026.26301); 48, 369--427 (1942; Zbl 0027.25301); 50, 1--94 (1944; Zbl 0063.07216)], with the metric defined by the invariant form \(ds^2 = dx^2 + dy^2\). lt is therefore to be expected that the isotropic geometry of the Airy surface \(z = z(x, y)\) yields the elasticity properties. That this is so is actually proved by showing that all invariant notions concerning a surface in isotropic geometry (like isotropic normal curvature, geodesic torsion, principal curvatures, mean curvature, curvature lines) do have a definite meaning in elasticity theory. The interesting link so put in evidence between this theory and isotropic geometry is apt to give further results.
Mathematics Subject Classification ID: 53Axx
zbMATH DE Number: 3174087
zbMATH Keywords: differential geometry in Euclidean spaces
zbMATH Keywords: Airy tension function
zbMATH Keywords: isotropic differential geometry
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3174087