M647 Spring 2017 Assignment 8, due Monday April 3 1. [10 pts] Suppose an object with mass m is launched at an angle from the ground, with initial height h0 and initial speed |~v0 | (see Figure 1). Use Newton's second law of motion to write down a system of ODE describing the flight of this object, and also write down appropriate initial conditions. Do not ignore air resistance. Note. Recall that we obtained an expression for force due to air resistance earlier in the semester, using dimensional analysis. ramp h0 x=0 Figure 1: Object in Problem 1. 2. [10 pts] Determine whether or not each of the following functions is Gateaux differentiable at ~x = 0. If the function is Gateaux differentiable, determine whether or not it is Frechet differentiable. a. ( x1 x2 (x1 , x2 ) 6= (0, 0) 2 2 f (x1 , x2 ) = x1 +x2 0 (x1 , x2 ) = (0, 0). b. x31 x21 +x22 (x1 , x2 ) 6= (0, 0) 0 (x1 , x2 ) = (0, 0). ( f (x1 , x2 ) = 3. [10 pts] Answer the following: a. We know from first-semester calculus that the length of the graph of a C 1 function y(t) connecting the point (0, ) to the point (T, ) is given by the arclength formula Z Tp L[y(t)] = 1 + y 0 (t)2 dt. 0 Compute L0 [y(t)], and use your result to show that the shortest such curve must be a straight line. Work in the function space y S := {y C 2 [0, T ] : y(0) = , y(T ) = }. b. [5 pts] As observed in class, the derivative L0 [y(t)] is a linear mapping on the space S0 := {h C 2 [0, T ] : h(0) = h(T ) = 0}. 1 We can compute the derivative of L0 [y(t)] (and so the second derivative of L[y(t)]) as the bilinear form L0 [y(t) + k(t)] L0 [y(t)] (h(t)). 0 L00 [y(t)](k(t), h(t)) := lim For the arclength functional from Part (a), compute L00 [y(t)], where y(t) is taken as the minimizer found in Part (a), and show that L00 [y(t)] is a positive definite bilinear form: that is, show that for any h S0 , h not identically 0, L00 [y(t)](h(t), h(t)) > 0. This is the variational second-order derivative condition for a minimizer, and ensures that y(t) is in fact a local minimizer. Note. This isn't necessary for the current problem, but FYI another way to look at this is as follows: we say L[y(t)] is twice differentiable if there exist a linear functional L0 [y(t)](h), a bilinear functional L00 [y(t)](h, h), and a functional \u000f[y(t)](h) so that 1 L[y + h] = L[y] + L0 [y](h) + L00 [y](h, h) + \u000f[y](h), 2 with \u000f[y](h) = 0, khk0 khk2 lim where k k denotes Holder norm (as discussed in class) on S0 . I.e., functionals have Taylor expansions. 4. [10 pts] Consider the system depicted in Figure 2 in which a pendulum with mass m2 is attached to a block with mass m1 that is attached to a wall by a spring with spring constant k. Take x = 0 to be the position of the block's center when the spring is neither stretched nor compressed, and denote the length of the pendulum by l. Write down the Lagrangian for this system and also the Euler-Lagrange equations. Ignore friction, which for practical purposes would make the system non-conservative. (The system is still conservative, but energy is being transferred into molecular motion.) 2 x k m1 unstretched l m2 Figure 2: Spring-pendulum system for Problem 4. 3