A particle of mass m is attracted to point O by the following central force: F(r) = -kr/r² where r is the position vector with respect to the point O, is its direction and r its magnitude. The particle has angular momentum L = rxp about O (conserved) and linear momentum p = mr (not conserved). The Runge-Lenzvector is defined by \underline{\epsilon}=\frac{\underline{p} \times \underline{L}}{m k}-\hat{r} A. Use the following vector identity to show that L² = r²p² - (rp)². Define the angle by \cos (\theta)=\frac{\underline{\epsilon} \cdot \underline{r}}{\epsilon r} . \text { where } \epsilon \text { denotes the magnitude of } \underline{\epsilon} \text {. Show that the following relation must hold between } r \text { and } \theta \text { : } r=\frac{l}{1+\epsilon \cos (\theta)}, \quad \text { where } \quad l \equiv \frac{L^{2}}{m k} C. Show that the magnitude of e is given by \epsilon=\sqrt{1+\frac{2 L^{2} E}{m k^{2}}}, \quad \text { where } \quad E=\frac{p^{2}}{2 m}-\frac{k}{r} hence, e is a constant of motion. D. Given the orbit in B, give a geometrical interpretation for the direction of the vector e as well as for its \text { magnitude, } \epsilon=\sqrt{\underline{\epsilon} \cdot \underline{\epsilon}} . (axb) (cxd) = (a · c)(b d) — (b.c)(a.d).