Geodesic flows on compact surfaces—as an application of Hamiltonian formalism

AbstractIn this paper, the author proves directly the equivalence between geodesic equations on a Riemannian manifold given by the Riemannian connection and the equations of motion of a unit-mass point outside of external forces on the manifold given by the priciples of least action (Section 1, Theorem 1). According to the Liouville theorem, the author gives the formulae of ratio of frequencies and periods of motion in two directions with which one can (1) distinguish periodicity or quasi-periodicity of the motion on a compact manifold determined by an integrable Hamiltonian system with two degrees of freedom; (2) obtain the period of the motion if it is periodic, if only the initial point of motion in phase space is known (Section 1, Theorem 2). As an application of Theorem 2, the geodesic flows on 2-torus and ellipsoid of three major axes of different lengths are studied (Section 2, Theorem 3; Section 3, Theorem 4). Two remarks (Section 2, Remark 1; Section 3, Remark 2) are made to discuss the extreme cases of the geodesic flows