Computation of caustics related to geometrical solutions with arbitrary initial data of eikonal equation

This study shows how to complete the support of caustics related to geometrical solutions (Lagrangian submanifolds) of the geometrical Cauchy problem for the eikonal equation, a special case of Hamilton-Jacobi equation. Although the computation is carried out for the simple Hamiltonian H(q, p) (1/2)p2 on Τ*R2, the authors can treat cases with arbitrary C2 initial data functions σ:Σ belonging to R, assigned on the initial manifold (curve) Σ immersed in R2. From the physical point of view, the caustics in such a way generated are closely linked to the caustics (in the sense of geometrical optics) determined by the curve Σ emitting light with intensity varying according to the differential of σ