Periodic orbits and non integrability of autonomous Glydén‐like systems

We apply the averaging theory of first order for studying the periodic orbits for an autonomous variation of the Gylden problem, i.e., a model of the planar two body problem where the gravitational constant undergoes small variations depending on the cartesian coordinate x. Of course this problem admits a Hamiltonian formulation. Two main results are shown.First, we show that at any negative energy level the Hamiltonian system has at least two periodic orbits. These periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy.Second, using these two families of periodic orbits we can prove the non integrability of the Hamiltonian system in the sense of Liouville‐Arnold, independently of the class of differentiability of the second first integral.Moreover the two tools that we use for proving our results, the first for studying periodic orbits and the second for studying the non Liouville‐Arnold integrability can be used for Hamiltonian systems with an ar...