Consider a particle with mass m that is moving under the influence of a central force described by the equation F=r3kr where k>0, and r represents the position vector of the particle at any given time t with respect to a suitable coordinate system and r represents the magnitude of r. (A) Show that the "Laplace-Runge-Lenz (LRL)" vector, given by A=pLmkrr is conserved for this particle. Here p and L represent respectively the momentum and angular momentum of the particle at any given time t. (B) Show that A satisfies the following two relations: AL=0andA=mk(1+mk22EL2)1/2 Here, E represents the total energy and, L and A represent respectively the magnitude of the angular momentum and "Laplace-Runge-Lenz (LRL)" vector of the particle. (C) Show that the orbit of the particle is described by conic sections r=1+cosr0 where is the angle between the vectors A and r and, r0=mkL2and=(1+mk22EL2)1/2 (D) Choose the angular momentum to be along the z-axis and the "Laplace-Runge-Lenz (LRL)" vector to be along the x-axis of a rectangular coordinate system and show that the locus of the momentum vector px2+(pyLA)2=(Lmk)2. is a circle of radius Lmk with center displaced from the origin at (0,LA)