$2$ Laplace$-$Runge$-$Lenz vector $(5$ marks$)$ For motion under a central conservative force, the total energy and the angular momentum L are both conserved. If the force obeys an inverse$-$square law force with potential energy V$($r$)=-$k$/$r$,$ the Laplace$-$Runge$-$Lenz vector, r A $=$ p $$ L $$ mk $,$ is also conserved. Show that A is perpendicular to L and, by recalling the standard orbit equation r l $=+1$ e cos $,$ where l $=$ L$2/$mk$,$ express the eccentricity e in terms of the length A of the Laplace$-$Runge$-$Lenz vector.