University of Zielona Góra

Relativistic Hamiltonian systems of $n$ degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential $V(\vq)$. In the limit of weak potential $2V(\vq)/mc^2\ll 1$, and smallmomentum $\abs{\vp}/ mc\ll 1$, these systems transform into the corresponding non-relativistic flat Hamiltonian's with the standard sum of kinetic energy plus potential $V(\vq)$. We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the H\'enon-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 $\vd$ satisfying $V'(\vd)=\gamma \vd$. 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 $\gamma^{-1}V''(\vd)$ are either 0, or 1. Certain integrable relativistic systems are presented.