Geodesics of the boundary in the obstacle problem and unfurled swallowtails
This paper is a study of the notion of a Lagrangian triad applied to an obstacle problem. A Lagrangian triad \((L, \ell, H)\) consists of a hypersurface \(H\) in a symplectic manifold \(M\), a Lagrangian submanifold \(L \subset M\), and a smooth hypersurface \(\ell \subset L\) such that \(L\) is tangent to \(H\) along \(\ell\). Lagrangian triads arise in the obstacle problem in the following way. Suppose \(\partial E\) in a Euclidean space \(E\) is a smooth hypersurface bounding an obstacle. The shortest path between two points in \(E\) lying on a straight line intersecting \(\partial E\) consists of two intervals of straight lines passing through the initial point and the end point and tangent to \(\partial E\), and an interval of a geodesic of \(\partial E\) which is tangent to these straight lines. A bundle of geodesics of \(\partial E\) and a family of rays tangent to them arises by varying the endpoint position. The bundle of geodesics of \(\partial E\) defines a Lagrangian manifold \(\ell' \subset T^* \partial E\). In \(T^* E\), take \(H\) to be the hypersurface of unit covectors, and consider the hypersurface \(\partial T^* E\) of covectors attached to points of \(\partial E\). The space of characteristics of the hypersurface \(\partial T^* E\) is naturally symplectomorphic to \(T^* \partial E\). Let \(L\) be the union of the characteristics of \(\partial T^*E\) mapped to \(\ell'\). If \(\ell = L \cap H\), then \((L, \ell, H)\) is a Lagrangian triad, and \(L\) is quadratically tangent to \(H\) along \(\ell\). The variety of characteristics of the hypersurface \(H\) passing through points of \(\ell\) is locally symplectomorphic to the unfurled swallowtail or to its suspension in an appropriate dimension. If a ray coming to the obstacle boundary is asymptotically tangent to \(\partial E\), then \(\ell'\) becomes singular along the geodesic of \(\partial E\) starting at the point of asymptotic tangency. The author studies the singularities of \(\ell'\) and determines that Lagrangian triads and unfurled swallowtails arise as normal varieties of geodesics of \(\partial E\).
scientific article; zbMATH DE number 859618