Intrinsic equations for a relaxed elastic line on an oriented surface
A regular arc on an oriented surface in 3-space is called a relaxed elastic line on that surface if it realizes an extremum of the total square (spatial) curvature among all arcs on the surface with the same initial point and the same initial direction. Extending work of the second author [Q. Appl. Math. 45, 515-527 (1987; Zbl 0633.73023)], the intrinsic equations for relaxed elastic lines on surfaces are derived. In a plane or on a sphere the relaxed elastic lines are given exactly by the geodesics. This property is shown to characterize parts of planes or spheres. On other surfaces there exist relaxed elastic lines which are not geodesics.
scientific article; zbMATH DE number 4056445