.; A particle of mass m moves in a central harmonic potential V(r) = >kr2 with a positive spring constant k. (a) Use the effective potential to show that all orbits are bounded and that E must exceed Emin = (kl2 /m)1/2. (b) Verify that the orbit is a closed ellipse with origin at the center. If the relation E/ Emin = cosh & defines the quantity , verify that the orbital parameters are given by a2 = el(mk)-1/2, b2 = e-$1(mk)-1/2, and 2 = 1 - e-25. Discuss the limiting cases E -> Emin and E >> Emin. (c) Prove that the period is 27(m/k)1/2, independent of E and l. Discuss this ele- mentary result