Catenaries in Riemannian surfaces (Q6566506)

The authors of this paper have worked in recent years on extending the concept of a ``catenary'' to the elliptic and hyperbolic plane as well as to simply isotropic space and de Sitter space. This paper features a unifying view on this subject and considers cateneries and more general \(\alpha\)-catenaries in abstract 2-dimensional Riemannian manifolds. In Euclidean space a catenary can be seen as a local minimizer of the gravitational potential energy with respect to a straight line. In Riemannian settings, the straight line is replaced by a geodesic and the distance used for computing the potential energy is the manifolds geodesic distance.\N\NCritical points of this variational problem are computed by standard techniques. Calculations are simplified by the use of ``semi-geodesic coordinates'' where one family of coordinate lines is unit speed geodesics emanating orthogonally from the reference geodesic. The authors provide several characterizations for catenaries and \(\alpha\)-catenaries in semi-geodesic coordinates and/or in terms of geodesic curvature. None of these results depends on the property that the reference curve is a geodesic, thus giving rise to the even more general concept of catenaries with respect to arbitrary curves.\N\NHaving set the theoretical and computational foundations, the article discusses and visualizes geodesics in several simple manifolds: Cylinders, cones, elliptic and hyperbolic plane, helicoid, surfaces of revolution (for which a Clairaut relation can be derived), and catenoid. The last section is dedicated to catenaries in the Grušin plane that allow particular simple parametrizations.