Mathematics – Dynamical Systems
Scientific paper
2011-12-14
Mathematics
Dynamical Systems
Accepted Physica D : Nonlinear Phenomena; 25 pages, 3 figures
Scientific paper
10.1016/j.physd.2011.11.010
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.
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