Chaos and integrability of relativistic homogeneous potentials in curved space

Abstract

Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential $$V(\varvec{q})$$ V ( q ) . In the limit of weak potential $$2V(\varvec{q})/mc^2\ll 1$$ 2 V ( q ) / m c 2 ≪ 1 , and small momentum $$|\varvec{p} |/ mc\ll 1$$ | p | / m c ≪ 1 , these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential $$V(\varvec{q})$$ V ( 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 $$\varvec{d}$$ d satisfying $$V'(\varvec{d})=\gamma \varvec{d}$$ V ′ ( d ) = γ d for a certain $$\gamma \ne 0$$ γ ≠ 0 . They are very strong: if the relativistic system is integrable in the Liouville sense, then either $$k=\pm 2$$ k = ± 2 , or all non-trivial eigenvalues of the re-scaled Hessian $$\gamma ^{-1}V''(\varvec{d})$$ γ - 1 V ′ ′ ( d ) are either 0, or 1. Certain integrable relativistic systems are presented.