of such an ordering does not restrict the architecture of the system. For example, the coordinates of an equal-mass binary with a circumbinary particle can be expressed in Jacobi coordinates. But note that the ordering might in general be non-unique and that it can change during an integration. This can have important implications for a numerical scheme using Jacobi coordinates.

The Jacobi coordinate $\mathbf{r}'_i$ of the $i$-th particles is the position relative to $\mathbf{R}_{i-1}$, the centre-of-mass of all the particles interior to the $i$-th particle: $$\begin{aligned}
\mathbf{r}'_i &=& \mathbf{r}_i - \mathbf{R}_{i-1},\quad\quad\quad\quad\text{for } \;i=1,\ldots,N-1\\
\text{where} \quad \mathbf{R}_{i} &=& \frac{1}{M_i}\sum_{j=0}^i m_j \mathbf{r}_j \quad \text{and } \quad M_{i} \;=\; \sum_{j=0}^i m_j.\end{aligned}$$ Other quantities such as the velocity and acceleration (also the coordinates in the variational equations, see below) transform in the same way. This is because the Jacobi coordinates are a linear function of the Cartesian coordinates, and the velocity is the time derivative of the position in both coordinates systems.

The momenta, however, transform differently[^1]. The momentum conjugate to $\mathbf{r}'_i$ and the corresponding Jacobi mass are given by $$\begin{aligned}
\mathbf{p}'_i = m'_i \dot{\mathbf{r}'}_i  = m'_i \mathbf{v'}_i \quad\quad\text{   and   } \quad\quad m'_i = m_i \frac{M_{i-1}}{M_{i}}=  \frac{m_i\,M_{i-1}}{m_i + M_{i-1}}.\label{eq:p}\end{aligned}$$ Note that the Jacobi mass 