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In this work, we develop a formalism to determine the geodesic orbits at a point associated with the surface of a rotating planet which at the same time is forced to move on a toroid geometric structure. Such an approach is based on a Lagrangian method applied to the geodesics on the torus. In particular, we consider a circular as well as elliptic toroids. We find parameters relating the angular velocity with the internal angular momentum (spin) of the planet. We show that this relation leads to a non-Newtonian potential. Thus, through our Lagrangian formalism we determine the constants of the motion, including the orbital angular momentum and the energy. 

geodesic, toroid, orbit, planets