Abstract
AbstractRelativistic 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.
Publisher
Springer Science and Business Media LLC