On the interior scattering of waves, defined by hyperbolic variational principles (Q913134)

This paper addresses the problem surrounding the transformation of waves of different kinds at interior points of a nonhomogeneous media. The waves are defined by linear hyperbolic variational systems. An essential review of contact geometry is included together with an important condition namely that the velocity of a moving contact element belongs to the hyperplane of the distribution iff the contact points velocity belongs to the moving contact element. The main result is that the light surface of a typical variational hyperbolic system is reduced in a neighborhood of a typical singular point, to the microlocal formal normal form, \(H=0\), where \(H=p^ 2_ 1\pm q^ 2_ 1-q^ 2_ 2\) for some local Darboux coordinates. That is to say coordinates such that the contact structure is defined by the equation, \(\alpha =0\), where \(\alpha =dz+(p dq-q dp)/2,\quad z\in {\mathbb{R}},\quad p\in {\mathbb{R}}^ D,\quad q\in {\mathbb{R}}^ D,\quad D>1.\) The proof is clearly developed in a series of twelve lemmas.