Bifurcations at the dawn of Modern Science

Pierre Coullet

doi:10.1016/j.crme.2012.10.026

Bifurcations at the dawn of Modern Science

Pierre Coullet ¹

¹ INLN, UMR 7335 CNRS – Université de Nice Sophia-Antipolis, 1361 routes des Lucioles, Sophia-Antipolis, 06560 Valbonne, France

Comptes Rendus. Mécanique, Volume 340 (2012) no. 11-12, pp. 777-788.

Résumé

In this article we review two classical bifurcation problems: the instability of an axisymmetric floating body studied by Archimedes, 2300 years ago and the multiplicity of images observed in curved mirrors, a problem which has been solved by Alhazen in the 11th century. We will first introduce these problems in trying to keep some of the flavor of the original analysis and then, we will show how they can be reduced to a question of extremal distances studied by Apollonius.

Publié le : 2012-11-01

DOI : 10.1016/j.crme.2012.10.026

Mots clés : Bifurcations, Singularities, Curved mirrors, Hydrostatics

Affiliations des auteurs :

Pierre Coullet ¹

¹ INLN, UMR 7335 CNRS – Université de Nice Sophia-Antipolis, 1361 routes des Lucioles, Sophia-Antipolis, 06560 Valbonne, France

@article{CRMECA_2012__340_11-12_777_0,
     author = {Pierre Coullet},
     title = {Bifurcations at the dawn of {Modern} {Science}},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {777--788},
     publisher = {Elsevier},
     volume = {340},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crme.2012.10.026},
     language = {en},
}

TY  - JOUR
AU  - Pierre Coullet
TI  - Bifurcations at the dawn of Modern Science
JO  - Comptes Rendus. Mécanique
PY  - 2012
SP  - 777
EP  - 788
VL  - 340
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crme.2012.10.026
LA  - en
ID  - CRMECA_2012__340_11-12_777_0
ER  -

%0 Journal Article
%A Pierre Coullet
%T Bifurcations at the dawn of Modern Science
%J Comptes Rendus. Mécanique
%D 2012
%P 777-788
%V 340
%N 11-12
%I Elsevier
%R 10.1016/j.crme.2012.10.026
%G en
%F CRMECA_2012__340_11-12_777_0

Pierre Coullet. Bifurcations at the dawn of Modern Science. Comptes Rendus. Mécanique, Volume 340 (2012) no. 11-12, pp. 777-788. doi : 10.1016/j.crme.2012.10.026. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.10.026/

Bibliographie
Cité par

[1] R. Levien The elastica: a mathematical history http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf

[2] K. Girstmair; G. Kirchner Expositiones Mathematicae, 26 (2008) no. 3, pp. 219-236 | arXiv

[3] T.L. Heath The Works of Archimedes, Dover Publications, Inc., 1897 (Reissue of the 1897 edition with the 1912 Supplement on The Method)

[4] Christiaan Huygens, Traité de la lumière, Leiden, 1690.

[5] R. Rashed Apollonius de Perge, Coniques: Livre V, Walter de Gruyter, Berlin, New York, 2008

[6] J.W. Bruce; P.J. Giblin Curves and Singularities, Cambridge University Press, 1992

[7] M.V. Berry; C. Upstill Progress in Optics, XVIII (1980), pp. 257-346

[8] J.D. Smith The remarkable Ibn al-Haytham, The Mathematical Gazette, Volume 76 (1992) no. 475, pp. 189-198

[9] R. Rashed Alhazen, un génie protéiforme au Moyen Age, Pour la science, vol. 315, 2004

[10] R. Rashed Apollonius de Perge, Coniques: Livre I, Walter de Gruyter, Berlin, New York, 2008

[11] R. Rashed Apollonius de Perge, Coniques: Livre II et III, Walter de Gruyter, Berlin, New York, 2010

[12] R. Rashed Apollonius de Perge, Coniques: Livre IV, Walter de Gruyter, Berlin, New York, 2010

[13] R. Rashed Apollonius de Perge, Coniques: Livre VI et VII, Walter de Gruyter, Berlin, New York, 2009

[14] P. Bouguer Traité du navire, et de ses mouvements, Jombert, Paris, 1746

[15] L. Euler Scientia Navalis, Oeuvres complètes série 2, oeuvres de mécanique et dʼastronomie, vols. 18, 19, Truesdell, Baltimore, 1967

[16] C. Dupin Applications la Géomètrie et de Mécanique, à la Marine, aux Ponts et Chaussées, etc … pour faire suite aux Développements de la Géomètrie, Bachelier, Paris, 1822

[17] A.I. Sabra Theory of Light: from Descartes to Newton, Cambridge University Press, 1981

[18] A. Ziggelaar How did the wave theory of light take shape in the mind of Christiaan Huygens?, Annals of Science, Volume 37 (1980) no. 2, pp. 179-187

[19] A. Quetelet Résumé dʼune nouvelle théorie des caustiques, Nouveaux mémoires de lʼAcadémie Royale des sciences et belles-lettres de Bruxelles, 1827

Cité par Sources :

Commentaires - Politique