Unit speed stationary points of the acceleration

Abstract

We consider the variational problem associated to the L2 norm of the angular acceleration for variations with constant length. This is a classical problem related to recent models for the spinning relativistic particles (both, massive and massless) and to Bernoulli’s elastica problem. Its interest arises also in computer graphics and in trajectory planning problems for rigid body motion. Explicit integration of the Euler-Lagrange equations is considerably difficult, in general. We show how one can use a geometric integration procedure to achieve this goal for unit speed critical points in real space forms. From the Euler-Lagrange equations, we obtain the geometric integration: critical curves are a family of ordinary helices in a real space form of dimension 3 at most. Moreover, ordinary helices in real space forms can be characterized in turn as geodesics in certain constant mean curvature flat “cylinders.” This will allow us to obtain the analytical integration of the Euler-Lagrange equations. We also determine the closed unit speed critical curves for this energy.