Symplectic singularities and optical diffraction
- 411 Downloads
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.
KeywordsCanonical Variety Symplectic Manifold Symplectic Structure Lagrangian Submanifolds Symplectic Geometry
Unable to display preview. Download preview PDF.
- Arnold, V.I. . Singularities in the variational calculus, Itogi Nauki, Contemporary Problems in Math. 22 3–55.Google Scholar
- Arnold, V.I. and Givental, A.B., Symplectic geometry . Itogi Nauki, Contemporary Problems in Math., Fundamental directions 4 5–139.Google Scholar
- Sommerfeld, A. . Thermodynamics and Statistical Mechanics, Academic Press, New York.Google Scholar
- Weinstein, A. . Lectures on symplectic manifolds, C.B.M.S. Conf. Series 29, Amer. Math. Soc., Providence.Google Scholar