This equation, like the eikonal equation, is an equation in the first partial derivatives and is of second degree. In terms of Maxwell’s equations light is understood as a wave obeying a wave equation. The eikonal is − and herein lies its main property − the generator of the equations for a contact transformation, through which any four determining data are coupled with each other, and which one might define to be the two conjugate lines σ and Σ. The eikonal equation (from German Eikonal, which is from Greek εἰκών, image) is a non-linear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory.It is derivable from Maxwell's equations of electromagnetics, and provides a link between physical (wave) optics and geometric (ray) optics. The eikonal approximation was not born in the study of quantum mechanics. The eikonal approximation in quantum mechanics works for processes involving the scattering of particles with large incoming momentum and when the scattering angle is very small. Lemma 2.1. Light we know obeys Maxwell’s equations. the application of eikonal approximation in high energy physics, especially in QCD. eikonal equation are known and one can find solutions of the coupling equation in explicit form. From the form of the eikonal equation there results a remarkable analogy between geometrical optics and the mechanics of material particles.

The motion of a material particle is determined by the Hamilton-Jacobi equation (16.11). 1.1 Example: Eikonal solution on a square Example 1.1. The relation between both methods is studyed in section IV. In section III we modify the method to solve the eikonal equation satisfying a given characteristic data at in nity. In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. In section II we review the method to give a general solution of the eikonal equation in flat space-time adapted to appropriate Cauchy data given at t= t 0 [4]. It originated far back in optics. Fo r a n m -dimensional Riemannian sp ace of c onstant curvature K with the metric ds 2 = 12.510 Lecture Notes 3.3.2007 The Wave Equation = ∇ 2 2 φ α φ, for a P wave • Often written 2 2 α φ φ ∇ − = 0 or L φ=( ) 0 where is an operator.L • Using d’Alembert’s Solution: φ = ( , ) ( ) x t A x e ⋅ − ωi k x t ( ), where the wave number k indicates the direction of the wave Ray Parameter and Slowness Consider the domain =[0;1]2 łR2 œ 1 2 u2 x+ 1 2 y= 1 in u=0 on =@: (1.2) The characteristics of (1.2) are given by x(s) =∇u(x(s)) with x(0) ∈@, and hence at any position in space we are traveling in the direction of maximal increase for u.