Integrability and chaos of relativistic Hamiltonian systems in 2D Riemannian manifolds

Abstract This paper is a continuation of our investigations of the dynamics and integrability of a relativistic particle moving under the influence of potential forces. Here, we examine particle motion in a general two-dimensional Riemannian manifold subjected to an external potential. In the limit of small momentum this system transforms into the corresponding non-relativistic system in a curved space extensively investigated in recent years. Our integrability results are formulated for relativistic systems with the metric and potential which are homogeneous functions of degrees n and k , respectively. First, we show that the integrability of a non-relativistic Hamiltonian is a necessary condition for the integrability of its relativistic version. Next, we derive the necessary integrability conditions from the analysis of the differential Galois groups of variational equations along two different particular solutions. These conditions, together with those for non-relativistic counterparts, give the final integrability obstructions. Obtained obstructions are used to explain integrability results for relativistic versions of well-known classical systems: the Elmandouh system with variable Gaussian curvature, the Hénon–Heiles-type systems, the position-dependent mass systems and the swinging Atwood machine. We compare the dynamics of the classical and corresponding relativistic versions systems in curved space using Poincaré sections and demonstrate that relativistic corrections frequently induce chaos. Moreover, we prove the non-integrability of these systems using the obtained necessary integrability conditions. Unexpectedly, we have also identified a new family of relativistic integrable Hamiltonian systems with position-dependent mass, which extends both classical integrable potentials and conditionally integrable systems defined at specific energy levels. Our results provide a deeper understanding of integrability and chaos in relativistic mechanics, which is currently of great importance for mathematical physics and astrophysics.