On the integrability of new examples of two-dimensional Hamiltonian systems in curved spaces

In this work, we inspect the integrability of a natural Hamiltonian system interpreted physically as the motion of a particle in the Euclidean plane under the effect of conservative forces derived from a certain type of a non-homogeneous potential. We announce the necessary conditions for its integrability by using the differential Galois theorem. We present three examples to clarify the applicability of the obtained results is easy and efficacious. Some of these examples restore the previous results in the literature, and one of them gives a new integrable case describing a generalization of the well-known Swinging Atwood machine. (C) 2020 Elsevier B.V. All rights reserved.