Elastica in SO(3)

Abstract

AbstractIn a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.