# Dynamical systems of null geodesics and solutions of Tomimatsu-Sato 2

## TL;DR

This paper investigates null geodesics and optical metrics in various spacetime solutions, especially Tomimatsu-Sato, revealing dualities and geometric properties through classical mechanics analogies.

## Contribution

It introduces a novel analysis of null geodesics as central force systems in spherically symmetric spacetimes, focusing on Tomimatsu-Sato and conformal dualities.

## Key findings

- Null geodesics modeled as central force problems.
- Derived Binet's equation for specific metrics.
- Identified conformal dualities preserving Jacobi metrics.

## Abstract

We have studied optical metrics via null geodesics and optical-mechanical formulation of classical mechanics, and described the geometry and optics of mechanical systems with drag dependent quadratically on velocity. Then we studied null geodesics as a central force system, deduced the related Binet's equation applied the analysis to other solutions of Einstein's equations in spherically symmetric spaces, paying special attention to the Tomimatsu-Sato metric. Finally, we examined the dualities between different systems arising from conformal transformations that preserve the Jacobi metric.