Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-20T00:26:36.169Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric Interpretation of Lagrangian Equivalence

Published online by Cambridge University Press:  20 November 2018

Shyuichi Izumiya
Shyuichi Izumiya*
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics, and wave front propagations are revealed.

Keywords

58K0557R4558K60wave front propagationsbig wave frontsgraph-like Legendrian unfoldingscaustics
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 806 - 812
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Arnol'd, V. I., Singularities of caustics and wave fronts. Mathematics and its Applications, 62, Kluwer Academic Publishers, Dordrecht 1990. http://dx.doi.Org/10.1007/978-94-011-3330-2 Google Scholar
[2] Arnol'd, V. I., Gusein-Zade, S. M., and Varchenko, A. N., Singularities of differentiate maps. vol. I. Mongraphs in Mathematics, 82, Birkhauser Boston, Boston, MA, 1985. http://dx.doi.Org/10.1007/978-1-4612-5154-5 Google Scholar
[3] Duistermaat, J. J., Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27(1974), 207–281. http://dx.doi.Org/!0.1OO2/cpa.3160270205 Google Scholar
[4] Hormander, L., Fourier integral operators!. Acta. Math. 127(1971), 79–183. http://dx.doi.Org/10.1007/BF02392052 Google Scholar
[5] Izumiya, S., Perestroikas of optical wave fronts and graphlike Legendrian unfoldings. J. Differential Geom. 38(1993), 485–500.Google Scholar
[6] Izumiya, S., Completely integrable holonomic systems of first-order differential equations. Proc. Royal Soc. Edinburgh Sect. A 125(1995), 567–586. http://dx.doi.Org/10.1017/S0308210500032686 Google Scholar
[7] Izumiya, S. and Takahashi, M., Spacelike parallels and evolutes in Minkowski pseudo-spheres. J. Geom. Phys. 57(2007), no. 8, 1569–1600. http://dx.doi.Org/10.1016/j.geomphys.2007.01.008 Google Scholar
[8] Izumiya, S. and Takahashi, M., Caustics and wave front propagations: Applications to differential geometry. Banach Center Publications, 82, Polish Acad. Sci. Inst. Math., Warsaw, 2008,125-142. http://dx.doi.Org/10.4064/bc82-0-9 Google Scholar
[9] Izumiya, S. and Takahashi, M., Pedal foliations and Gauss maps of hypersurfaces in Euclidean space. J. Singul. 6(2012), 84–97. http://dx.doi.Org/10.5427/jsing.2012.6g Google Scholar
[10] Nye, J. F., Natural focusing and fine structure of light. Insitute of Physics Publishing, Bristol, 1999. http://dx.doi.Org/10.1006/abbi.1999.1180 Google Scholar
[11] Zakalyukin, V. M., Reconstructions of fronts and caustics depending one parameter and versality of mappings. (Russian) Current problems in mathematics, 22, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983. pp. 56–93.Google Scholar
[12] Zakalyukin, V. M., Envelope of families of wave fronts and control theory. Proc. Steklov Inst. Math. 209(1995), 114–123.Google Scholar