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The classical theory of attractive central force problem on the standard(flat) Euclidean plane can be generalized to surfaces by reformulating thebasic underlying physical principles by means of differential geometry.Attractive central force problems on state manifolds appear quite often and inseveral different context ranging from nonlinear control theory to mobilerobotics, thermodynamics, artificial intelligence, signal transmission andprocessing and so on.  The aim of the present paper is to analyze the variational properties of thecircular periodic orbits in the case of attractive power-law potentials of theRiemannian distance on revolution's surfaces. We compute the stabilityproperties and the Morse index by developing a suitable intersection index inthe Lagrangian Grassmannian and symplectic context.

Morse index of circular solutions for attractive central force problems on surfaces