Conservation Laws

In the absence of non-gravitational forces and external torques, total energy and angular momentum are conserved. Often this fact can be used to simplify a problem. The following formulae assume the origin is at the center of mass (so bulk motion can be ignored).

Total Energy: $E = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 - \frac{Gm_1m_2}{\vert\mathb... ...1\vert} = \frac{1}{2} \mu v^2 - \frac{Gm_1m_2}{r} = \frac{1}{2} \mu v_\infty^2$

Angular Momentum: $\mathbf{L} = m_1 (\mathbf{r}_1 \mbox{\boldmath$\times$}\mathbf{v}_1) + m_2 (\m... ... (\mathbf{r} \mbox{\boldmath$\times$} \mathbf{v}) = \mu v_q q = \mu v_\infty b$

( $\mu = m_1 m_2/M$ ; = impact parameter = $r \sin \phi$ )

Total Energy:	$E = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 - \frac{Gm_1m_2}{\vert\mathb... ...1\vert} = \frac{1}{2} \mu v^2 - \frac{Gm_1m_2}{r} = \frac{1}{2} \mu v_\infty^2$
Angular Momentum:	$\mathbf{L} = m_1 (\mathbf{r}_1 \mbox{\boldmath$\times$}\mathbf{v}_1) + m_2 (\m... ... (\mathbf{r} \mbox{\boldmath$\times$} \mathbf{v}) = \mu v_q q = \mu v_\infty b$