The behavior of geodesic curves on even seemingly simple surfaces can be
surprisingly complex. In this paper we use the Hamiltonian formulation of the
geodesic equations to analyze their integrability properties. In particular, we
examine the behavior of geodesics on surfaces defined by the spherical
harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to
prove that the geodesic equations on all surfaces defined by the sectoral
harmonics are not integrable, and we use Poincar\'{e} sections to demonstrate
the breakdown of regular motion.Comment: Accepted Physica D : Nonlinear Phenomena; 25 pages, 3 figure
