In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm may be described as follows: We choose two of the N bodies with position coordinates xj and xk and we replace them with one virtual body at their centre of mass. We define the relative position coordinate rjk = xjxk.

We then repeat the process with the N1 bodies consisting of the other N2 plus the new virtual body. After N1 such steps we will have Jacobi coordinates consisting of the relative positions and one coordinate giving the position of the last defined centre of mass.

For the N-body problem the result is:

with

is the center of mass is the relative coordinate between the particles 1 and 2:

The result one is left with is thus a system of N, from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian.

If one is interested in evaluating a free energy operator in these coordinates, one obtains

In the calculations can be useful the following identity

1. .

References