first term, $\mathcal{H}_0$, simply describes the motion of the centre-of-mass $\mathbf{r}'_0$ along a straight line. For that reason this term is often ignored. However, we keep it which will allow us to integrate particles without any restriction to a particular frame of reference.

The terms $\mathcal{H}_{\rm Kepler}$ can be split up further into a sum of $$\begin{aligned}
\left(\mathcal{H}_{\rm Kepler}\right)_i &=& \frac{{\mathbf{p}'_i}^2}{2m'_i} - \frac{G m'_i M_{i} }{|\mathbf{r}'_i|}. \label{eq:HKepler}\end{aligned}$$ Each of the Hamiltonians $\left(\mathcal{H}_{\rm Kepler}\right)_i$ describes the Keplerian motion of the $i$-th particle with mass $m_i$ around the centre-of-mass of all interior particles with total mass $M_{i-1}$.

After some more algebra, the interaction term can be simplified and split into two parts, one of which can be easily computed in Jacobi coordinates and the other in Cartesian coordinates of the inertial frame $$\begin{aligned}
\operatorname{\mathcal{H}}_{\rm Interaction} = 
\sum_{i=2}^{N-1} \frac{ G{m'}_i M_i }{|\mathbf{r}'_i|} 
 - \sum_{i=0}^{N-1} \sum_{\substack{j=i+1\\j\neq 1}}^{N-1} \frac{Gm_im_j}{|\mathbf{r}_i-\mathbf{r}_j|}.\end{aligned}$$

One important point to note is that our choice of $\operatorname{\mathcal{H}}_\pm$ is slightly different from that used by [@solarsystemdynamics] and [@WisdomHolman1991]. These authors use $$\begin{aligned}
\left(\operatorname{\mathcal{H}}_\pm\right)_{\rm WH1991} &=& \sum_{i=1}^{N-1}\frac{G m'_i \, \mathcal{M}_i }{|\mathbf{r}'_i|}.\end{aligned}$$ where $\mathcal{M}_i = m_0 \frac{M_i}{M_{i-1}}$. Their choice leads to the usual disturbing function in perturbation theory. We conducted various tests but found no 