# Conservation laws for classical particles in Anti-de Sitter-Beltrami space

## TL;DR

This paper derives conservation laws for classical particles in Anti-de Sitter space, transforming between different coordinate systems and limits, including nonrelativistic cases, to understand particle dynamics in curved spacetime.

## Contribution

It introduces a method to construct conserved quantities for particles in AdS space using Beltrami coordinates and explores their nonrelativistic limits.

## Key findings

- Conserved quantities are expressed in Beltrami coordinates.
- Limits to Lorentz-Fock spacetime are obtained for small velocities.
- Energy expressions for nonrelativistic particles in AdS are derived.

## Abstract

In this paper we consider the conservation laws for classical particles in $AdS_4$. At first we parameterize a geodesic line and construct conserved quantities with analog of five dimensional Minkowski space-time $M^{(2,3)}$. Consequently we change $AdS_4$ space to AdS-Beltrami space-time and write conserved quantities in the Beltrami coordinates. Finally we take a limit for small velocity $\dot{x}\ll c$ and we get the conserved quantities in Lorentz-Fock space-time. And finally we give out the energy of the nonrelativistic noncosmological particle under the cosmological limit.