Math 642-600 Assignments — Spring 2022

Assignment 1 - Due Monday, 2/9/2022.

• Read section 5.1.
• Do the following problems.
  1. Show that the functional $J(y):=\int_0^1( xy^2+y'^2)dx$ is Fréchet differentiable, if the the $C^1[0,1]$ norm is used.
  2. Problem 2, page 204 (§ 5.1).
  3. Problem 7, page 204 (§ 5.1).
  4. Hanging chain problem. A chain having uniform linear density ρ hangs between the points (0,0) and (1,0). The total mass m, which is fixed, and the total energy E of the chain are
     
     Assuming that the chain hangs in a shape that minimizes the energy, find the shape of the hanging chain.
  5. Consider the Brachistochrone problems that we did in class.
     1. In the case where $y(\theta_0)=B$, find the time $T$ it takes to go from $(0,0)$ to $(1,B)$ in terms of $\theta_0$ and $B$. (Of course, $\theta_0$ is a function of $B$.)
     2. Show that the solution to the brachistocrone problem with natural boundary conditions is the same as the one obtained using $B=2/\pi$. Find the value of $T$ when a natural boundary condition is given at $x=1$.
  6. Consider the functional J(y) = ∫_a^bF(x,y,y′)dx, where y ∈ C¹ and y(a)=A, y(b)=B are fixed.
     1. Let $f\in C[a,b]$ and suppose that for every $\eta\in C^{1}[a,b]$ such that $\eta(a)=0$ and $\eta(b)=0$, $\int_a^b f(x)\eta'(x)dx=0$. Show that $f$ is contant on $[a,b]$.
     2. Let y extremize J, subject to all of the conditions above. Verify that if η ∈ C¹ and η(a)=η(b)=0, then
        Δ J = ∫_a^b (-g(x) + F_y′(x,y,y′))η′(x)dx, where g(x) = ∫_a^xF_y(u,y,y′)du.
        
     3. Use (a) and (b) to show that -g+F_y′(x,y,y′)=c, a constant. From this, it follows that F_y′(x,y,y′)∈ C¹. (Du Bois-Reymond, 1879. The point is that one need not assume that y′′ exists.)
  7. Smoothing spline. Let J[y] = ∫₀¹ (y′(x))²dx. Let the admissible set for J be all piecewise C¹ curves on [0,1] that satisfy y(k/n) = y_k , for k = 0, 1, 2, ..., n, with the discontinuities in y′ appearing only at the points x_k = k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(x_k,y_k), k = 0, ..., n}. (Hint: y(x) is in C¹[x_k, x_k+1]).)

Assignment 2 - Due Friday, 3/4/2022.

• Read sections 5.2 and 5.4 and my notes on the Courant-Fischer Theorem.
• Do the following problems.
  1. Problem 10, page 204 (§ 5.1).
  2. Problem 11, page 204 (§ 5.1).
  3. Problem 6, page 205 (§ 5.2).
  4. Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a diagram.)
  5. Consider the torus (surface of a "doughnut") obtained by rotating a circle with center $(R,0,0)$ and radius $r < R $ about the $z$ axis. Find the metric tensor for it and use this tensor to obtain the 2nd order differential equations for the geodesics.
  6. Find the Legendre transformation H(p) for F(x) = x^T A x, where x is in Rⁿ and A is a symmetric, positive definite n×n matrix.
  7. 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.

Assignment 3 - Due Friday, 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 $b_n = \frac{2}{n}J_n(\varepsilon)$, $n = 1, 2,\cdots$, 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).

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