Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T11:31:03.188Z Has data issue: false hasContentIssue false
  • English
  • Français

On Legendre transformations and elementary catastrophes

Published online by Cambridge University Press:  24 October 2008

M. J. Sewell
M. J. Sewell
Affiliation:
Department of Mathematics, University of Reading
Get access Rights & Permissions [Opens in a new window]

Extract

The purpose of these remarks is to use elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and certain elementary catastrophes. Multi-valued Legendre transforms appear in subjects such as non-linear elasticity and non-convex optimization.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 147 - 163
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Budiansky, B.Theory of buckling and post-buckling of elastic structures. Advances in Applied Mechanics 14 (1974), 1–65.Google Scholar
(2)Isnard, C. A. and Zeeman, E. C. Some models from catastrophe theory in the social sciences. In Use of models in the social sciences, ed. Collins, L. (London, Tavistock, 1974).Google Scholar
(3)Koiter, W. T. On the stability of elastic equilibrium. Nasa Technical Translation F– 10, 833, 1967. (Translation of a 1945 Thesis in Dutch.)Google Scholar
(4)Koiter, W. T. Post-buckling analysis of a simple two-bar frame. In Recent progress in applied mechanics (The Folke Odgvist Volume), pp. 337–354 (Wiley, 1966).Google Scholar
(5)Koiter, W. T. On the complementary energy theorem in non-linear elasticity theory. In Trends in applications of pure mathematics to mechanics, ed. Fichera, G., pp. 207–232. (London, Pitman, 1976).Google Scholar
(6)Ogden, R. W.Inequalities associated with the inversion of elastic stress-deformation relations and their implications. Math. Proc. Cambridge Philos. Soc. 81 (1977), 313–324.Google Scholar
(7)Sewell, M. J.On the connexion between stability and the shape of the equilibrium surface. J. Mech. Phys. Solids 14 (1966), 203–230.Google Scholar
(8)Sewell, M. J.Some mechanical examples of catastrophe theory. Bulletin of the Institute of Mathematics and its Applications 12 (1976), 163–172.Google Scholar
(9)Thom, R.Structural stability and morphogenesis (Reading, Mass., W. A. Benjamin, Inc., 1975).Google Scholar
(10)Woodcock, A. E. R. and Poston, T.A geometrical study of the elementary catastrophes (Berlin, Springer-Verlag, 1974).Google Scholar