[3] The planar motion of a mass point under the attraction of a central source is modelled by the system :e = 401% (011 iv} = f(r)g, where r := V212 + ya. (0.2) (a) [4 marks] Change the equations to polar coordinates s: = reos 6, y = rsin 6, (1",3) E (O, 00) x 11% to obtain the equivalent system 0 = r612 f0") (0.3) 2m + 125? = 0,. (0.4) (Show all your calculations.) Verify that the equation (0.4) above is equivalent to the fact that along any solution (11:), 90)) we have (1 dt ( (060)):0 r=r (t) 9:19\" and so the function c(t) := r2(t)9(t) is constant along the motion, that is r2(t)(t) = constant 2 r3199. This statement is written short-hand r29 = c where the constant c is xed by the initial conditions. ()h [1 mark] Prove that if c = 0 the motion is linear, that is 9(3) = constant = 9(0) for all t. c)[3 marks] Substituting 9 = into equation (0. 3) one obtains 1:2 r: :'2 f0\") (0.5) Writing the above (0.5) as an ODE system with r' = 1.1, prove that the function E(r,v):= 'u 2++j s)ds is constant along any solution (r(t),11(t))