Math 642-600 Assignments — Spring 2016

Assignment 1 - Due Friday, 1/29/2016.

• 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) = ∫abF(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 = ∫ab (-g(x) + Fy′(x,y,y′))η′(x)dx, where g(x) = ∫axFy(u,y,y′)du.
2. Use (a) to show that -g+Fy′(x,y,y′)=c, a constant. From this, it follows that Fy′(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) = yk , for k = 0, 1, 2, ..., n, with the discontinuities in y′ appearing only at the points xk = k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(xk,yk), k = 0, ..., n}. (Hint: y(x) is in C1[xk, xk+1]).)
6. Let x = (x,y) and let J[x] = x2y(x2 + y2)− 1 if x0, J[0] = 0. Show that the Gâteaux derivative exists at at x = 0 for every direction η = (h,k). Also, show that the Fréchet derivative doesn't exist at x = 0.

Assignment 2 - Due Friday, 2/5/2016.

• Do the following problems.
1. Problem 5, page 204 (§ 5.1).
2. Problem 9, page 204 (§ 5.1).
3. Problem 12, page 204 (§ 5.1).
4. 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.
5. Problem 6, page 205 (§ 5.2).
6. Find the Legendre transformation H(p) for F(x) = xT A x, where x is in Rn and A is a symmetric, positive definite n×n matrix.

Assignment 3 - Due Friday, 2/12/2016.

• Read sections 6.1 and 6.2.
• 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. The Hamiltonian of a system having a Lagrangian $L(q_1,q_2,\ldots,q_n, \dot q_1,\dot q_2,\ldots,\dot q_n)$ is defined as the Legendre transform of $L$ with respect to the $\dot q_j$'s. (See pg. 187). In class we showed that, for a planet (mass $m$) in orbit around the sun (mass=$M$), where the constant $K=GmM$, the polar form of the Lagrangian is $L=\frac{m}{2}\big(\dot r^2 + r^2 \dot \theta^2\big)+ Kr^{-1}$.
1. Show that the Hamiltonian $H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - Kr^{-1}$.
2. Show that the angular momentum $p_\theta=\ell$ is a constant of the motion.
3. Show that the Hamiltonian is a constant of the motion, $H=E$, where $E$ is the total enery of the system.
4. Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell \frac{d}{d\theta}(r^{-1})$.
5. Let $u = r^{-1}- mK\ell^{-2}$. Show that $2m E\ell^{-2} + m^2K^2\ell^{-4} = \big(\frac{du}{d\theta}\big)^2 + u^2$. Solve this first order differential equation for u in terms of $\theta$.
6. 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, 2/19/2016.

• Read sections 6.1 and 6.2.
• Do the following problems.
1. Consider a mass-spring system that has two masses, m1 and m2, and three springs, with constants k1, k2 and k3. The system is placed horizontally on a frictionless surface. The first spring is attached to a support on the left and to the first mass on its right. The second mass is attached to the first mass on its left and to the second on its right. The third spring is attached to the second mass on its left and to a support on its right. The masses are at displacements x_1 and x2 from their equilibrium positions. Find the Hamiltonian of the system.
2. Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
3. Problem 4, page 208 (§ 5.4). Use the function $\phi_1(x) = x(1-x)/\sqrt{30}$ for the approximate eigenfunction for lowest eigenvalue, $\lambda_1$. (We did this in class.) To approximate the next eigenvalue, take $\phi_2$ to be a cubic that is orthogonal to $\phi_1$ and $\phi_2\|_{L^2[0,1]}=1$. Calculate the approximation to $\lambda_2$ using $\phi_2$.
4. 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?
5. 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.

Assignment 5 - Due Wednesday, 3/2/2016.

• Read sections 6.2 and 6.4.
• Do the following problems.
1. Do problems 4, 5 and 11 in the Exercises for Complex Variables
2. Find all of the Laurent expansions about $z=0$ for $f(z) = \frac{1}{(z^2-1)(z+2)}$
3. Section 6.1: 4, 6
4. Section 6.2: 6, 9, 10
5. Section 6.4: 2, 4, 9
6. Let $\xi \ge 0$. Find $\hat f(\xi) := \frac{1}{2\pi}\int_{-\infty}^\infty \frac{e^{i\xi x}}{1+x^4}dx$.

Assignment 6 - Due Wednesday, 3/30/2016.

• Read sections 7.1 and 7.2.
• Do the following problems.
1. 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 $Y_0(z) =\frac{2}{\pi} \big(\gamma+\log\big(\frac{z}{2}\big)\big)J_0(z) - \frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^n}{(n!)^2}\big(\frac{z}{2}\big)^{2n}\big(\sum_{k=1}^n\frac{1}{k}\big).$
2. Problem 2, page 279 (§ 6.5).
3. Problem 8(b) page 280 (§ 6.5).
4. Problem 9, page 280 (§ 6.5). (Do $J_{3/2}(z)$.)
5. 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 − tp), where tp 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 $u = \tau + \sum_{n=1}^\infty (2/n)J_n(n\varepsilon) \sin(n\tau).$

Assignment 7 - Due Wednesday, 4/13/2016.

• Do the following problems.
1. Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find the norm and adjoint $L^\ast$. You may use any of the problems below.

2. Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L \in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the exterior of the disk $\{\lambda \in {\mathbb C} \colon |\lambda| \le ||L||\}$. (Consequently, $\sigma(L)$ is contained in the disk.)

3. Let $K \in {\mathcal B}(\mathcal H)$ and let $L:D \to \mathcal H$ be a densely defined, closed linear operator.
1. Show that $L+K$ is defined on $D$ and is also closed.
2. Show that $(L+K)^\ast$ is defined on the domain of $L^\ast$ and that $(L+K)^\ast = L^\ast + K^\ast$.
3. Suppose that $0 \in \rho(L)$. Show that $0 \in \rho(L^\ast)$ and that $(L^{-1})^\ast = (L^\ast)^{-1}$.

4. Let $L = L^\ast$ be defined on a domain $D$. In class, we showed that $\sigma_r(L) = \emptyset$ and that $\sigma_d(L) \subset \mathbb R$. Show that $\sigma_c(L) \subseteq \mathbb R$. Thus, $\sigma(L)$ is a subset of the reals.

5. Let $L = L^\ast$ be defined on a domain $D$ and suppose that $L$ satisfies $\langle Lf,f \rangle \ge 0 \$, for all $f\in D$. Show that $\sigma(L) \subseteq [0, \infty)$.

Assignment 8 - Due Friday, 4/22/2015.

• Do the following problems.
1. Let $L$ be a closed, densely defined operator on a Hilbert space $\mathcal H$. Prove the following.
1. If $\lambda$ and $\mu$ belong to $\rho(L)$, then $R_\lambda-R_\mu=(\lambda-\mu)R_\lambda R_\mu$.
2. $R_\lambda R_\mu=R_\mu R_\lambda$.
3. Let $D_L$ be the domain of $L$. Then, on $D_L$, $LR_\lambda = R_\lambda L$.

2. Let $L=L^\ast$ and let $C$ be a positively oriented, simple closed curve in the resolvent set, encircling $\sigma_0\subset \sigma(L)\in \mathbb R$ — but no other points in $\sigma(L)$. (Note: the analyticity of $R_z$ implies that $C$ may be deformed into any contour with the same properties.) Let $P_C := -\frac{1}{2\pi i}\oint_C (L-zI)^{-1}dz = -\frac{1}{2\pi i}\oint_C R_z dz$.
1. Consider the curve $C'$ that is the complex conjugate of $C$. Show that $C'$ contains $\sigma_0$, that it is negatively oriented, and that $P_{-C'} =-\frac{1}{2\pi i}\oint_{-C'} R_z dz$.
2. Use part (a) to show that $P_C^\ast = P_C$.
3. Let $C_1$ and $C_2$ both encircle $\sigma_0$. In addition, suppose that they never intersect each other. Use problem 1(a) and an interchange of integrals to show that to show that $P_C^2=P_C$, so $P_C$ is an orthogonal projection.

3. Let $L=A$ be a self-adjoint, $n\times n$ matrix. Suppose that $A$ has $r$ distinct eigenvalues, arranged in the order $\lambda_1 < \lambda_2 < \cdots < \lambda_r$. In addition, let the orthogonal projections onto the eigenspaces be $Q_1,\ldots,Q_r$. Suppose $a < b$, and neither is an eigenvalue of $A$; let $C_\varepsilon$ be a positively oriented rectangular contour, with corners $a-i\varepsilon$, $b-i\varepsilon$, $b+i\varepsilon$, and $a+i\varepsilon$. Show that $\lim_{\varepsilon\downarrow 0}-\frac{1}{2\pi i} \oint_{C_\varepsilon} (A-zI)^{-1}dz = E_b-E_a,$ where $E_\lambda :=\sum_{j=1}^k Q_j= \sum_{j=1}^k H(\lambda - \lambda_j)Q_j$, where $H(t)$ is the Heaviside step function.

4. Section 7.2: 1(b).

Assignment 9 - Not to be handed in.

• Suggested problems
1. Prove the following theorems:
1. If $f,g \in L^1(\mathbb R) \cap L^2(\mathbb R)$, then $f\ast g\in L^1(\mathbb R) \cap L^\infty(\mathbb R)$.
2. If $f\in L^1(\mathbb R)$, then its Fourier transform $\hat f$ is in $C_0(\mathbb R)$.

2. Consider the one dimensional heat equation, $u_t = u_{xx}$, with $u(x,0) = f(x)$, where $-\infty < x < \infty$ and $0 < t < \infty$. You are given that ${\mathcal F}[e^{-x^2}] =\sqrt{2\pi}e^{-\xi^2/4}$. By taking the Fourier transform in $x$, show that the solution $u(x,t)$ is given by $u(x,t) = \int_{\mathbb R}K(x-y,t)f(y)dy,\ K(\xi,t) = (4\pi t)^{-1/2} e^{-\xi^2/4t}.$ (The function $K(\xi,t)$ is the one-dimensional heat kernel.)

3. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and eibxT(x).

4. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^{(m)}(x)| \le c_m(1+t^2)^{n_m}$ for all nonnegative integers m. Here $c_m$ and $n_m$ depend on $g$ and $m$. Show that $fg$ is in Schwartz space. Use this result to explain how to define the product $g(x)T(x)$, where $T$ is a tempered distribution.

5. Section 10.3: 1, 3, 14.

Updated 5/3/2016 (fjn)