Math 642-600 Assignments — Spring 2019

Assignment 1 - Due Wednesday, 1/23/19

• Read section 5.1.
• Do the following problems.
  1. Problem 2, page 204 (§ 5.1).
  2. Problem 7, page 204 (§ 5.1).
  3. 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.
  4. 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 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.
        
     2. Use (a) 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.)
  5. 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 Monday, 2/4/2019.

• Do the following problems.
  1. Problem 5, page 204 (§ 5.1).
  2. Problem 9, page 204 (§ 5.1).
  3. Problem 11, page 204 (§ 5.1).
  4. 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.
  5. The metric of a surface in $\mathbb R^3$ having parametric form $x=x(u,v),y=y(u,v), z=z(u,v)$ is defined by $ds^2=E(u,v)du^2 + 2F(u,v)dudv +G(u,v)dv^2$. 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 for it and use it to obtain the 2nd order differential equations for the geodesics.
  6. Problem 6, page 205 (§ 5.2).
  7. 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.

Assignment 3 - Due Monday, 2/18/2019.

• Do the following problems.
  1. Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a diagram.)
  2. Problem 8, page 206 (§ 5.2). (Note: There is an error in the expression for L. The (dθ/dt)² sin²θ) should be replaced by (dφ/dt)² sin²θ.)
  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.

Assignment 4 - Due Friday, 3/1/2019.

• Do the following problems.
  1. Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
  2. Use a quadratic polynomial to estimate the lowest eigenvalue for the Sturm-Liouville problem with Lu = − u′′ = λu, u(0) = 0, u′(1)+u(1) = 0.
  3. Problem 6, page 208 (§ 5.4). Take Ω to be a 2D disk centered at 0 and having radius r = a. How does the lowest eigenvalue change with a? Explain.
  4. Consider $f(z)=1/z$, which is analytic, and therefore continuous, in $\mathbb C \setminus \{0\}$. Show that $f$ has no analytic anti-derivative in $\mathbb C \setminus \{0\}$. Does this contradict Morera's Theorem? Explain.
  5. Show that the coefficients for a Laurent series valid in a given annulus are unique.
  6. Find all of the Laurent expansions about $z=0$ for $f(z) = \frac{1}{(z^2-1)(z+3)}$.

• Read section 6.5.1, 6.5.2 and 7.1.
• Do the following problems.
  1. Section 6.4: 3, 7 (assume $a$ and $k$ are positive), 9, 20, 25.

  2. Let $\xi \in \mathbb R$ and consider the integral $G(\xi)=\int_{-\infty}^\infty e^{-x^2}e^{i\xi x}dx$. Show that $G(\xi)= \sqrt{\pi}e^{-\xi^2/4}$, given that $G(0)=\sqrt{\pi}$.

  3. Show that $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt$ is analytic for $\text{Re}(z)>0$.

Updated 3/8/2019 (fjn)