§1
Softened gravitational force
\[F \;=\; \htmlClass{tw-G}{G}\,\frac{m_1\,m_2}{\bigl(\,r^{2} + \htmlClass{tw-eps}{\varepsilon}^{2}\,\bigr)^{\htmlClass{tw-n}{n}/2}}\]
§2
Velocity-Verlet integration
\[r_{\text{new}} = r + v\,\htmlClass{tw-dt}{\Delta t} + \tfrac{1}{2}\,a\,\htmlClass{tw-dt}{\Delta t}^{2}\]
\[v_{\text{new}} = v + \tfrac{1}{2}\,(a + a_{\text{new}})\,\htmlClass{tw-dt}{\Delta t}\]
\[E \;=\; \tfrac{1}{2}\sum m\,v^{2} \;-\; \htmlClass{tw-G}{G}\sum_{i\,\lt\,j}\frac{m_i\,m_j}{r_{ij}}\]