Nonexistence of the final first integral in the Zipoy-Voorhees space-time

Abstract

We show that the geodesic motion in the Zipoy-Voorhees space-time is not Liouville integrable, in that there does not exist an additional first integral meromorphic in the phase-space variables.

Comment on “Nonexistence of the final first integral in the Zipoy-Voorhees space-time”

“…

The accuracy of the numerical findings of [1], regarding the existence of additional first integrals in the Zipoy-Voorhees space-time, was recently questioned [2]. In this comment, it is shown that the discrepancy between the results of [1] and [2] is not due to issues related to numerical accuracy, as claimed in [2], but due to different choice of coordinates used in [2].

…”

“…5 of [1] and Fig. 1 of [2] as claimed in [2], and in any case one should be very careful to discern any difference by just inspecting a surface of section like those in [1,2]. Furthermore, the detailed self-similar structures of the islands of stability shown on the surface of section of [1] is a qualitative criterion which excludes the possibility of "artificial" chaos produced by an inaccurate numerical scheme.…”

Non-integrability of geodesic dynamics of Chazy-Curzon space-time

“…Geodesic motion on Zipoy-Voorhees space-times known as a γ-metrics, has been studied via numerical methods, in [15] and [16] and analytically in [13] and [17]. The Zipoy-Voorhees metrics with a parameter δ have as special cases both Minkowski flat space-time (δ = 0) and Schwarzschild (δ = 1) space-time.…”

On integrability of certain rank 2 sub-Riemannian structures

“…The technique we use in sections 5, 6 and 7 is inherited from the work [KM1], where it was exploited to prove rigorously non-existence of lowdegree integrals for the Zipoy-Voorhees metric from general relativity (for related work on this topic, see [LG,MPS,V1]). We will explain the method in detail in Sections 3 and 4.…”