Math 642-600 Current Assignment — Spring 2020

Assignment 3 - Wednesday, 3/18/2020.

• Do the following problems.
  1. Problem 1, page 207 (§ 5.4). (Hint: Use Courant Fischer. Be very careful about what the admissible functions are.)

  2. Problems 3, 6, 9, 23, 25, pages 277-278 (§ 6.4).

  3. For a planet having mass $m$ in orbit around the sun (mass=$M$), the potential is $V(r)=-mMG/r$, where $G$ is the gravitational constant. In class, we showed that the Hamiltonian for this system is $H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - mMGr^{-1}$, and that $E=H$ and $\ell=p_\theta$ are constants of motion for the system. Inserting these constants in $H$, we have $E=\frac{1}{2m}\big(p_r^2+r^{-2}\ell^2\big)-mMGr^{-1}$.
     1. Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell \frac{d}{d\theta}(r^{-1})$, if we change the independent variable from $t$ to $\theta$.

     2. Let $u = r^{-1}- m^2MG\ell^{-2}$. Show that $2m E\ell^{-2} + m^4M^2G^2\ell^{-4} = \big(\frac{du}{d\theta}\big)^2 + u^2$. To simplify notation, let $\rho=(m^2MG\ell^{-2})^{-1}$ and $\gamma=2m\rho^2 E \ell^{-2}$. This puts the equation above in the form $u=r^{-1} -\rho^{-1}$ and $\rho^{-2}(1+\gamma)=\big(\frac{du}{d\theta} \big)^2 + u^2$. (Obviously, $\gamma+1>0$. For the orbit to be closed, one needs $\gamma <0$. Consequently, $0<\gamma+1<1$.) Solve this differential equation for $u$ in terms of $\theta$ and obtain $r=r(\theta)$.

     3. Use your solution to the previous part to establish Kepler's laws of planetary motion:
        1. The orbit of a planet is an ellipse, with the sun at one focus.
        2. The radius vector from the sun to the planet sweeps out equal areas in equal times.
        3. The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. The proportionality constant is the same for all planets.

Updated 2/28/2020(fjn)