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,^[3] and in celestial mechanics.^[4] An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees.^[5] In words, the algorithm is described as follows:^[5]

Let m_j and m_k be the masses of two bodies that are replaced by a new body of virtual mass M = m_j + m_k. The position coordinates x_j and x_k are replaced by their relative position r_jk = x_j − x_k and by the vector to their center of mass R_jk = (m_j q_j + m_kq_k)/(m_j + m_k). The node in the binary tree corresponding to the virtual body has m_j as its right child and m_k as its left child. The order of children indicates the relative coordinate points from x_k to x_j. Repeat the above step for N − 1 bodies, that is, the N − 2 original bodies plus the new virtual body.

For the N-body problem the result is:^[2]

${\displaystyle {\boldsymbol {r))_{j}={\frac {1}{m_{0j))}\sum _{k=1}^{j}m_{k}{\boldsymbol {x))_{k}\ -\ {\boldsymbol {x))_{j+1}\ ,\quad j\in \{1,2,\dots ,N-1\))$

${\displaystyle {\boldsymbol {r))_{N}={\frac {1}{m_{0N))}\sum _{k=1}^{N}m_{k}{\boldsymbol {x))_{k}\ ,}$

with

${\displaystyle m_{0j}=\sum _{k=1}^{j}\ m_{k}\ .}$

The vector ${\displaystyle {\boldsymbol {r))_{N))$ is the center of mass of all the bodies and ${\displaystyle {\boldsymbol {r))_{1))$ is the relative coordinate between the particles 1 and 2:

The result one is left with is thus a system of N-1 translationally invariant coordinates ${\displaystyle {\boldsymbol {r))_{1},\dots ,{\boldsymbol {r))_{N-1))$ and a center of mass coordinate ${\displaystyle {\boldsymbol {r))_{N))$, from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian equal to ${\displaystyle 1}$.

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

${\displaystyle H_{0}=-\sum _{j=1}^{N}{\frac {\hbar ^{2)){2m_{j))}\,\partial _((\boldsymbol {x))_{j))^{2}=-{\frac {\hbar ^{2)){2m_{0N))}\,\partial _((\boldsymbol {r))_{N))^{2}\!-{\frac {\hbar ^{2)){2))\sum _{j=1}^{N-1}\!\left({\frac {1}{m_{j+1))}+{\frac {1}{m_{0j))}\right)\partial _((\boldsymbol {r))_{j))^{2))$

In the calculations can be useful the following identity

${\displaystyle \sum _{k=j+1}^{N}{\frac {m_{k)){m_{0k}m_{0k-1))}={\frac {1}{m_{0j))}-{\frac {1}{m_{0N))))$.