Singularities of Caustics and Wave Fronts

ISBN-13: 9781402003332 / Angielski / Miękka / 2001 / 259 str.

V. Arnold; Vladimir I. Arnol'd

Opis

One service mathematics has rendered the 'Et moi, ... ) si j'avait su comment en revenir, human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. ErieT. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Recenzje

`Anyone who is interested in the mathematics of geometric optics, singularities bifurcation theory or just in applications of global analysis may find this as one of the few `must have' books.'
Mathematics, 734 (1991)

Spis treści

1 Symplectic geometry.- 1.1 Symplectic manifolds.- 1.2 Submanifolds of symplectic manifolds.- 1.3 Lagrangian manifolds, fibrations, mappings, and singularities.- 2 Applications of the theory of Lagrangian singularities.- 2.1 Oscillatory integrals.- 2.2 Lattice points.- 2.3 Perestroikas of caustics.- 2.4 Perestroikas of optical caustics.- 2.5 Shock wave singularities and perestroikas of Maxwell sets.- 3 Contact geometry.- 3.1 Wave fronts.- 3.2 Singularities of fronts.- 3.3 Perestroikas of fronts.- 4 Convolution of invariants, and period maps.- 4.1 Vector fields tangent to fronts.- 4.2 Linearised convolution of invariants.- 4.3 Period maps.- 4.4 Intersection forms of period maps.- 4.5 Poisson structures.- 4.6 Principal period maps.- 5 Lagrangian and Legendre topology.- 5.1 Lagrangian and Legendre cobordism.- 5.2 Lagrangian and Legendre characteristic classes.- 5.3 Topology of complex discriminants.- 5.4 Functions with mild singularities.- 5.5 Global properties of singularities.- 5.6 Topology of Lagrangian inclusions.- 6 Projections of surfaces, and singularities of apparent contours.- 6.1 Singularities of projections from a surface to the plane.- 6.2 Singularities of projections of complete intersections.- 6.3 Geometry of bifurcation diagrams.- 7 Obstacle problem.- 7.1 Asymptotic rays in symplectic geometry.- 7.2 Contact geometry of pairs of hypersurfaces.- 7.3 Unfurled swallowtails.- 7.4 Symplectic triads.- 7.5 Contact triads.- 7.6 Hypericosahedral singularity.- 7.7 Normal forms of singularities in the obstacle problem.- 8 Transformation of waves defined by hyperbolic variational principles.- 8.1 Hyperbolic systems and their light hypersurfaces.- 8.2 Singularities of light hypersurfaces of variational systems.- 8.3 Contact normal forms of singularities of quadratic cones.- 8.4 Singularities of ray systems and wave fronts at nonstrict hyperbolic points.