Hamilton-Jacobi equation in momentum space

Abstract

The application of the Hamilton-Jacobi equation to isotropic optical materials leads to the well-known eikonal equation which provides the surfaces normal to the ray trajectories. The symmetry between the coordinates x=(x(1),x(2),x(3)) and the momenta p=(p(1),p(2),p(3)) in the Hamiltonian formulation of Geometrical Optics establishes a dual Hamilton-Jacobi equation for "wavefronts" in the momentum space. This equation is also an eikonal equation when the refractive index distribution has spherical symmetry. In this case, another spherical symmetric refractive index distribution may exist such that the ray trajectories in the coordinates and momentum space are exchanged (examples of this case are given: Maxwell fish-eye, Eaton lens and Luneburg lens). The relationship between the wavefronts in the coordinate and momentum space is also analyzed. Curved orthogonal coordinates are considered as well.