In modern optics, light can be described at different levels: as rays, as scalar waves, as vector fields, and as quantum fields. In the first three levels, there are singularities—characteristic features, useful in interpreting phenomena at that level. In geometrical optics, the singularities are ray caustics; in scalar wave optics, they are phase singularities (=wave dislocations= wave vortices = nodal manifolds); in vector waves, they are singularities where the polarisation of light is purely linear or purely circular. The singularities at each level are dissolved at the next level. Similar singularities occur in all waves, not just light.

Newtonian forces depending only on position but which are non-conservative, i.e. whose curl is not zero, are termed ‘curl forces’. They are non-dissipative, but cannot be generated by a Hamiltonian of the familiar isotropic kinetic energy + scalar potential type. Nevertheless, a large class of such non-conservative forces (though not all) can be generated from Hamiltonians of a special type, in which kinetic energy is an anisotropic quadratic function of momentum. Examples include all linear curl forces, some azimuthal and radial forces, and some shear forces. Included are forces exerted on electrons in semiconductors, and on small particles by monochromatic light near an optical vortex. Curl forces imply restrictions on the geometry of periodic orbits, and non-conservation of Poincaré's integral invariant. Some fundamental questions remain, for example: how does curl dynamics generated by a Hamiltonian differ from dynamics under curl forces that are not Hamiltonian?

The force on a particle with complex electric polarizability is known to be not derivable from a potential, so its curl is non-zero. This ‘curl force’ is studied in detail for motion near an anisotropic optical vortex of arbitrary strength. Fundamental questions are raised by the fact that although the curl force requires the polarizability to have a non-zero imaginary part, reflecting absorption or scattering (‘dissipation’) in the internal dipole dynamics, the particle motion that it generates is non-dissipative (volume-preserving in the position-velocity state space).

Optical phenomena visible to everyone have been central to the development of, and abundantly illustrate, important concepts in science and mathematics. The phenomena considered from this viewpoint are rainbows, sparkling reflections on water, mirages, green flashes, earthlight on the moon, glories, daylight, crystals and the squint moon. And the concepts involved include refraction, caustics (focal singularities of ray optics), wave interference, numerical experiments, mathematical asymptotics, dispersion, complex angular momentum (Regge poles), polarisation singularities, Hamilton’s conical intersections of eigenvalues (‘Dirac points’), geometric phases and visual illusions.