Two-dimensional geodesic flows having first integrals of higher degree

Abstract

We present a family of riemannian metrics on two-sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2. They also have the property that all geodesics are closed. (2000): 53C22, 58F05 Mathematics Subject Classification

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

“…4. The integrable metrics shown to exist by Kiyohara in [7] are Zoll metrics, which means that all the geodesics are closed. This suggests the conjecture that the corresponding systems are in fact SI.…”

“…More recently Tsiganov gave a new example of this kind [12] on the two-sphere. In fact Kiyohara [7] was the first to show the existence 1 of integrable systems with globally defined riemannian metrics having integrals of arbitrary integer degree which, furthermore, are Zoll metrics.…”

Topology and Integrability in Lagrangian Mechanics

“…Beyond degree 4, Kiyohara has provided a construction of a smooth Riemannian metric H with an independent first integral F of degree k for any k≥1. I n this construction, the metric H depends on a functional modulus, and so for each k, the set is infinite dimensional [35].…”

“…Without this additional hypothesis, there is very little known. Indeed, the extremely valuable construction of Kiyohara is the only construction that provides a smooth Riemannian Hamiltonian with a polynomial-in-momenta first integral of degree N > 3-super-integrable or not [35,38].…”

On a family of integrable systems on S2 with a cubic integral of motion

“…Recall that in [5,6] all the geodesic flows are defined in implicit form only (see also discussion in [4]). …”