# Physical meaning of the conserved quantities on anti-de Sitter geodesics

## TL;DR

This paper explores the conserved quantities governing geodesic motion in anti-de Sitter spacetime, revealing two SO(3) vectors analogous to the Runge-Lenz vector that shape the geodesic trajectories.

## Contribution

It introduces the identification of two SO(3) vectors as conserved quantities that determine the shape of geodesics in anti-de Sitter space, akin to the Runge-Lenz vector in classical mechanics.

## Key findings

- Two conserved SO(3) vectors determine geodesic shapes.
- Geodesic trajectories are characterized by ten conserved quantities.
- The role of these vectors parallels classical orbital mechanics.

## Abstract

The geodesic motion on anti-de Sitter spacetimes is studied pointing out how the trajectories are determined by the ten independent conserved quantities associated to the specific SO(2,3) isometries of these manifolds. The new result is that there are two conserved SO(3) vectors which play the same role as the Runge-Lenz vector of the Kepler problem, determining the major and minor semiaxes of the ellipsoidal anti-de Sitter geodesics.