COMPUTATION OF CAUSTICS RELATED TO GEOMETRICAL SOLUTIONS WITH ARBITRARY INITIAL DATA OF THE EIKONAL EQUATION,

We show how to use the Projective Duality to compute the support of caustics related to geometrical solutions (i.e. Lagrangian submanifolds) of the geometrical Cauchy problem for the eikonal equation, a special case of the Hamilton-Jacobi equation. Although the computation is carried out for the simple Hamiltonian function $H(q,p)=\frac{1}{2} p^2$ on $T^*\mathbb{R}^2$, we will deal with {\em arbitary} $C^2$ initial data $\sigma: \Sigma \to \mathbb{R}$, assigned on the initial manifold (curve) $\Sigma$ embedded in $\mathbb{R}^2$.