Math 642-600 Current Assignment — Spring 2022

Assignment 3 - Due Wednesday, 4/13/2022.

• Read sections 7.1 and 7.2.
• Do the following problems.
  1. Consider a time-independent change of variables from $(q,p)$ to $(Q,P)$, i.e., $Q_j=Q_j(q_1,\ldots,q_n,p_1,\ldots,p_n)$ and $P_j=P_j(q_1,\ldots,q_n,p_1,\ldots,p_n)$. Using any of the four generating for a time independent Hamiltonian, the new Hamiltonian is obtained from the old by a simple subsitution $K(Q,P)=H(q(Q,P),p(Q,P))$.
     1. Using any of the generating functions, show that $\sum_{j=1}^n \dot q_j p_j- \sum_{j=1}^n \dot Q_j P_j=\frac{dG}{dt}$, where $G$ is a function of the various coordinates, independent of the Hamiltonian.

     2. Show that the differential form $\omega=\sum_{j=1}^n p_j dq_j- \sum_{j=1}^n P_jdQ_j$ is exact; i.e., $d\omega=0$. Conversely, if $\omega$ is exact, then the transform is canonical.

  2. Use the previous problem to show that $Q=\log(1+q^{1/2}\cos(p)), \ P= 2(1+q^{1/2}\cos(p))q^{1/2}\sin(p)$ is a canonical transformation. Show that the generating function $F_3(p,Q)=-(e^Q-1)^2\tan(p)$. (Note one can use $\widetilde\omega=pdq+QdP=d\omega+d(PQ)$ instead of $\omega$.)

  3. Show that, for $n\ge1$, $ \frac{\Gamma'(n+1+z)}{\Gamma(n+1+z)} = \sum_{k=1}^n\frac{1}{k+z} + \frac{\Gamma'(1+z)}{\Gamma(1+z)} $. Specifically, for $z=0$, $\frac{\Gamma'(n+1)}{\Gamma(n+1)} = \sum_{k=1}^n\frac{1}{k}- \gamma$, where $\gamma=-\Gamma'(1)$ is the Euler-Mascheroni constant. (One can show that $\gamma:= \lim_{n\to \infty}\big(\sum_{j=1}^n\frac{1}{j} - \log(n)\big) \approx 0.5772$.) Use this formula to obtain \[ Y_0(z) =\frac{2}{\pi} \big(\gamma+\log\big(\frac{z}{2}\big)\big)J_0(z) - \frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^n}{(n!)^2}\big(\frac{z}{2}\big)^{2n} \big(\sum_{k=1}^n\frac{1}{k}\big). \]
  4. A planet moving around the Sun in an elliptical orbit, with eccentricity 0 < ε < 1 and period P, has time and angle (position) related in the following way. Let τ = (2π/P)(t − t_p), where t_p is the time when the planet is at perihelion -- i.e., it is nearest the Sun. Let θ be the usual polar angle and let u be an angle related to θ via \[ (1 - \varepsilon)^{1/2} \tan(u/2) = (1 + \varepsilon)^{1/2} \tan(\theta/2). \] It turns out that τ = u − ε sin(u). All three variables θ, u, and τ are measured in radians. They are called the true, eccentric, and mean anomalies, respectively. (Anomaly is another word for angle.)
     1. Show that one may uniquely solve τ = u − ε sin(u) for u = u(τ), that u is an odd function of τ, and that g(τ) = u(τ) − τ is a 2π periodic function of τ.
     2. Because g is odd and 2π periodic, it can be represented by a Fourier sine series, \[ g(\tau) = \sum_{n=1}^\infty b_n \sin(n\tau). \] Show that $b_n = \frac{2}{n}J_n(n\varepsilon)$, $n = 1, 2,\cdots$, where $J_n$ is the $n^{th}$ order Bessel function of the first kind. Thus, we have that $ u = \tau + \sum_{n=1}^\infty (2/n)J_n(n\varepsilon) \sin(n\tau). $

  5. Problem 2(a,b), page 328 (§ 7.1).

Updated 4/7/2022 (fjn)