Symplectic singularities and optical diffraction

  • S. Janeczko
  • Ian Stewart
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1463)


Singularities of symplectic mappings are important in mathematical physics; for example in optics they determine the geometry of caustics. Here we survey the structure of symplectic singularities and extend the results from mappings to symplectic relations, by making use of Lagrangian varieties (which may have singularities) in place of Lagrangian manifolds. We explain how these ideas apply to classical ray-optical diffraction: the highly singular geometry in physical space turns out to be the projection of well-behaved geometry in phase space. In particular we classify generic caustics by diffraction in a half-line aperture, and discuss diffraction at a circular obstacle.


Canonical Variety Symplectic Manifold Symplectic Structure Lagrangian Submanifolds Symplectic Geometry 
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  1. Abraham, R. and Marsden, J.E.[1978]. Foundations of Mechanics, (2nd ed.), Benjamin, Reading.zbMATHGoogle Scholar
  2. Alexander, S.B., Berg, I.D. and Bishop, R.L. [1987]. The Riemannian obstacle problem, Illinois J. Math. 31 167–184.MathSciNetzbMATHGoogle Scholar
  3. Arnold, V.I. [1981]. Lagrangian manifolds with singularities, asymptotic rays, and the open swallowtail, Funct. Anal. Appl. 15 235–246.CrossRefGoogle Scholar
  4. Arnold, V.I. [1983]. Singularities in the variational calculus, Itogi Nauki, Contemporary Problems in Math. 22 3–55.Google Scholar
  5. Arnold, V.I. and Givental, A.B., Symplectic geometry [1985]. Itogi Nauki, Contemporary Problems in Math., Fundamental directions 4 5–139.Google Scholar
  6. Arnold, V.I., Gusein-Zade, S.M, and Varchenko, A.N. [1985]. Singularities of Differentiable Maps vol I, Birkhäuser, Boston.CrossRefzbMATHGoogle Scholar
  7. Berry, M.V.[1981]. Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ‘billiard’, Eur. J. Phys. 2 91–102.MathSciNetCrossRefGoogle Scholar
  8. Chernoff, P.R. and Marsden, J.E. [1974]. Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Math. 425, Springer, Berlin.CrossRefzbMATHGoogle Scholar
  9. Dangelmayr, G. and Güttinger, W. [1982]. Topological approach to remote sensing, Geophys. J.R. Astr. Soc. 71 79–126.CrossRefzbMATHGoogle Scholar
  10. Dirac, P.A.M. [1950]. Generalized Hamiltonian Dynamics, Canad. J. Math. 2 129–148.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Guillemin, V. and Sternberg, S. [1984]. Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge.zbMATHGoogle Scholar
  12. Givental, A.B. [1988]. Singular Lagrangian manifolds and their Lagrangian mappings, Itogi Nauki, Contemporary Problems in Mathematics 33 55–112.MathSciNetzbMATHGoogle Scholar
  13. Golubitsky, M., Schaeffer, D.G. [1979]. A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 21–98.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Janeczko, S. [1986]. Generating families for images of Lagrangian submanifolds and open swallowtails, Math. Proc. Camb. Phil. Soc. 100 91–107.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Janeczko, S. [1987]. Singularities in the geometry of an obstacle, Suppl. ai Rend. del Circolo Matematico di Palermo (2nd ser.) 16 71–84.MathSciNetzbMATHGoogle Scholar
  16. Keller, J.B. [1978]. Rays, waves and asymptotics, Bull. Amer. Math. Soc. 84 727–749.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Luneburg, R.K. [1964]. Mathematical Theory of Optics, Univ. of California Press, Berkeley.zbMATHGoogle Scholar
  18. Poston, T. and Stewart, I. [1978]. Catastrophe Theory and its Applications, Pitman, London.zbMATHGoogle Scholar
  19. Rychlik, M.R. [1989]. Periodic points of the billiard ball map in a convex domain, J. Diff. Geom. 30 191–205.MathSciNetzbMATHGoogle Scholar
  20. Scherbak, O.P. [1988]. Wave fronts and reflection groups, Uspekhi Mat. Nauk 43 125–160.MathSciNetGoogle Scholar
  21. Sniatycki, J., and Tulczyjew, W.M. [1972]. Generating forms of Lagrangian submanifolds, Indiana Math. J. 22 267–275.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Sommerfeld, A. [1964]. Thermodynamics and Statistical Mechanics, Academic Press, New York.Google Scholar
  23. Tulczyjew, W.M. [1974]. Hamiltonian systems, Lagrangian systems and the Legendre transformation, Symposia Mathematica, 14 247–258.MathSciNetzbMATHGoogle Scholar
  24. Weinstein, A. [1978]. Lectures on symplectic manifolds, C.B.M.S. Conf. Series 29, Amer. Math. Soc., Providence.Google Scholar
  25. Weinstein, A. [1981]. Symplectic geometry, Bull. Amer. Math. Soc. 5 1–13.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Zakalyukin, V.M. [1976]. On Lagrangian and Legendrian singularities, Funct. Anal. Appl. 10 23–31.CrossRefGoogle Scholar

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© Springer-Verlag 1991

Authors and Affiliations

  • S. Janeczko
  • Ian Stewart

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