ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 6

(February 9, 2011)

Read Landau-Lifshitz §15. Due February 11.

(18) Consider the effective potential of the Kepler problem

U_{eff}(r) = −\frac{α}{r} + \frac{L^{2}}{2 μ r^{2}} .

(A) Calculate the value r_{0}^{min} where U_{eff}(r) has its minimum.

(B) Calculate U_{eff}^{min} = U_{eff}(r_{0}^{min}).

(C) Assume an energy E < 0 and calculate the values of the turning points r_{min} and r_{max}.

Due in class, 4 points.

(19) Assume 0 < e < 1 for the eccentricity and transform the elliptic equation

\frac{p}{r} = 1 + e cos(ϕ) , p > 0 into the form \frac{x^{′2}}{a^{2}} + \frac{y′2}{b^{2}} = 1 .

This means, derive the definitions of x′, y′, major half-axis a and minor half-axis b in terms of x, y, p and e. Due in class, 4 points.

(20) Use the inertial frame of §15 of the book and plot the orbits for the four intial conditions of the table in set 5 (best in one figure). Indicate the initial positions. Due February 18 before class, 12 points.