Non-Integrability of Geodesic dynamics of Chazy-Curzon space-time

Abstract

We study the integrability of the geodesic equations of the Chazy- Curzon space-time. It was established that for the equilibrium point $p_{\rho }=p_z=z=0$ and, ${\rho }_0\in (1,2)$, there are only periodic solutions, the Hamiltonian system, describing geodesic motion of Chazy-Curzon space-time has no additional analytic first integral. Our approach is based on the following: if the system has a family of periodic solutions around an equilibrium and if the period function is infinitely branched then the system has no additional analytical first integral.

References in corpus (4)

• The non-integrability of the Zipoy-Voorhees metric
• Spacetime Encodings II - Pictures of Integrability
• Reducibility of Valence-3 Killing Tensors in Weyl's Class of Stationary and Axially Symmetric Space-Times
• Geodesic dynamics in Chazy-Curzon spacetimes