# Geodesic dynamics in Chazy-Curzon spacetimes

## TL;DR

This paper numerically analyzes geodesic motion in Chazy-Curzon spacetimes, finding exclusively regular trajectories that imply the existence of Carter's constant, and observes oscillatory motions similar to the MacMillan problem.

## Contribution

It provides the first thorough numerical study of geodesics in Chazy-Curzon metrics, revealing regular motion and potential conserved quantities.

## Key findings

- All geodesics exhibit regular motion.
- Evidence suggests the existence of Carter's constant.
- Oscillatory motions similar to the MacMillan problem are observed.

## Abstract

In the last decades, the dynamical studies around compact objects became a subject of active research, partially motivated by the observed differences in the profiles of the gravitational waves depending on the dynamics of the system. In this work, via the Poincar\'e section method, we conduct a thorough numerical analysis of the dynamical behavior of geodesics around Chazy-Curzon metrics. As the main result, we find only regular motions for the geodesics in all cases, which suggest the existence of the so-called Carter's constant in this kind of exact solutions. Moreover, our simulations indicate that in the two-particle Chazy-Curzon solution, some oscillatory motions take place as in the classical MacMillan problem.