18.03 Problem Set 6.5

This is another set of problems not to be handed in, just to give you practice for the Hour Exam on Wednesday, April 21.

Syllabus

23. (F 2 Apr) Complex or repeated eigenvalues: Notes LS 1.3, 1.4.

24. (M 5 Apr) Dynamics of linear autonomous systems: Notes GS 1–5, EP 7.3 (522– 530), Portrait Gallery Handout.

25. (W 7 Apr) Dynamics of nonlinear autonomous systems; linearization: Notes GS 6–7.

26. (F 9 Apr) Example: the nonlinear pendulum: EP 7.5 (561).

27. (M 12 Apr) Initial Value problems: Notes LS 2.1–2.3, EP 5.7 (443–447).

28. (W 14 Apr) Matrix exponentials: LS 2.4, EP 5.8 (454–457; the series expression for the exponential is not essential).

29. (F 16 Apr) Spinning books—a review: Handout on Euler’s equations.

30. (W 21 Apr) Hour Exam III

Part I.

27. (M 12 Apr) EP 5.7: 15, 17, 19.

28. (W 14 Apr) EP 5.8: 1, 3; Notes LS 32, 36.

29. (F 16 Apr) TBA

Part II.

27. (M 12 Apr) Suppose ẍ + e^{x} = 0. Write down the associated system of first order equations, using y = ẋ. Then find the “first integral” of this autonomous system, by thinking of y as a function of x, verifying that yy′ = ẏ (where y′ = dy/dx, ẏ = dy/dt), and solving the resulting first order ODE for y as a function of x. You have the Matlab tools by now to cause it to plot the resulting functions y(x) and to plot the trajectories of the system, and you may feel like checking your work.

28. (W 14 Apr) (a) Verify that multiplication by the complex number a + bi is

represented by the matrix A(a + bi) = [ \begin{matrix}a & −b \\ b & a \end{matrix} ], in the sense that if (a + bi)(x + yi) =

p + qi then A(a + bi) [ \begin{matrix}x \\ y \end{matrix} ] = [ \begin{matrix}p \\ q \end{matrix} ]. Check that A(z)A(w) = A(zw) and A(z + w) =

A(z) + A(w). Convince yourself that A(cos θ + i sin θ) = [ \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ] has the

effect of rotating counterclockwise by an angle θ, and that (for r real) A(r) = [ \begin{matrix}r & 0 \\ 0 & r \end{matrix} ] has the effect of multiplying vectors by r. What are the eigenvalues of A(z)? What are the eigenvectors? What is e^{A(z)}? Does this help convince you that the expression e^{a+bi} = e^{a}(cos b + i sin b) is reasonable?

(b) Answer Notes LS 37, by the following method: the first column of e^{At} is the solution

to the IVP ẋ = Ax with initial condition x(0) = [ \begin{matrix}1 \\ 0 \end{matrix} ], and the second column is the

solution with initial condition x(0) = [ \begin{matrix}0 \\ 1 \end{matrix} ]. Find these solutions.

29. (F 16 Apr) Use ode45 and plot to get a better idea of how the axis moves according to the Euler equations ((7) in the handout on them) with λ_{1} = 1, λ_{2} = 10, λ_{3} = 100 and initial conditions l_{1}(0) = 1, l_{2}(0) = 2, l_{3}(0) = 3. Take the time interval to be 20. You might want to see what the trajectory swept out by the axis of rotation in three-space looks like, using plot3 (about which you can find some information using help plot3).