# On a Center-of-Mass System of Coordinates for Symmetric Classical and Quantum Many-Body Problems

## TL;DR

This paper introduces a new coordinate system for symmetric many-body problems that includes the center-of-mass and preserves permutation symmetry, offering advantages over traditional Jacobi coordinates.

## Contribution

It constructs and proves the uniqueness of a coordinate system that maintains permutation symmetry and includes the center-of-mass, applicable to classical and quantum many-body problems.

## Key findings

- Provides a new coordinate system preserving permutation symmetry.
- Demonstrates applications to classical and quantum problems.
- Generalizes to systems with different species of bodies.

## Abstract

In the context of classical or quantum many-body problems involving identical bodies, a linear change of coordinates can be constructed with the properties that it includes the center-of-mass as one of the new coordinates and preserves the inherent permutation symmetry of both the Hamiltonian and the admissible states. This has advantages over the usual system of Jacobi coordinates in the study of many-body problems for which permutation symmetry of the bodies plays an important role. This paper contains the details of the construction of this system and the proof that these properties uniquely determine it, up to trivial modifications. Examples of applications to both classical and quantum problems are explored, including a generalization to problems involving groups of different species of bodies.