Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential \(V(\varvec{q})\) . In the limit of weak potential \(2V(\varvec{q})/mc^2\ll 1\) , and small momentum \(|\varvec{p} |/ mc\ll 1\) , these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential \(V(\varvec{q})\) . We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector \(\varvec{d}\) satisfying \(V'(\varvec{d})=\gamma \varvec{d}\) for a certain \(\gamma \ne 0\) . They are very strong: if the relativistic system is integrable in the Liouville sense, then either \(k=\pm 2\) , or all non-trivial eigenvalues of the re-scaled Hessian \(\gamma ^{-1}V''(\varvec{d})\) are either 0, or 1. Certain integrable relativistic systems are presented.
