Precession. Starting from Eq. (8.41), use Mathematica2 to find the equation of motion of a particle in a gravitational field. (Hint: use NDSolve.) Use PolarPlot to graph the orbit in polar coordinates.3)

(a) Plot the orbits for several values of c ?l2/?? (? is the force constant) on the same graph. [Hint: the Show command combines multiple graphs.] (b) Check analytically that the circular orbit occurs for the expected value of c. (c) For a single value of c, investigate the effect of deviations from New- ton's Law of Gravitation. Plot an eccentric orbit for F ?r?2.1 and for F ? r?1.9. (Hint: selecting ? between 0 and 6? gives a nice looking graph.)

u () =-1(0) - F (8.41) 1 2 1 () 2