Assume a particle moves in two-dimension (r, 0) in a central potential given by V (r) = ar2. (a) Find the Lagrangian L of the system (b) Find the equations of movement and the constant of motion (c) Reduce the two-dimensional problem to a one-dimensional problem by finding the effective potential Veff (r) (d) Show that using ac E = ac r + 0 - C you obtain E = mr2/2 + Ven (r), and also (10 points) 2 2E - V(r) L2 m m 2r2 (e) Use the transformations dr dr de do dt and y = 1/r2 and obtain (10 points) 1 dy 2 2mEy - 2/ 2 2ma 2 do = L2 L2 (f) complete square, integrate and show what are the conics solution for the differential equation above