Curves whose curvature depends on their position and null curves

Abstract

We show that, apart from degeneracies, determining a plane curve whose curvature depends on its position essentially consists of determining a null curve in the Lorentzian 3-space when the null tangent direction depends on its position. We use this point of view to investigate the intrinsic equations for the n-elastic curves. We show how the problem of prescribed null tangent direction in terms of the position can be solved by quadratures when the prescription exhibits sufficient symmetries. This problem is formalized in terms of a convenient contact 3-form.