Chapter 5.4, Problem 36E

Rigid Body Nutation. Euler’s equations describe the motion of the principal-axis components of the angular velocity of a freely rotating rigid body (such as a space station), as seen by an observer rotating with the body (the astronauts, for example). This motion is called nutation. If the angular velocity components are denoted by $x,y$, and $z$, then an example of Euler’s equation is the three-dimensional autonomous system

$\begin{array}{rll}dx/dt& =& yz,\\ dy/dt& =& -2xz,\\ dz/dt& =& xy.\end{array}$

The trajectory of a solution $x\left(t\right),\text{}y\left(t\right),z\left(t\right)$ to these equations is the curve generated by the points $\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)$ in $xyz$-phase space as $t$ varies over an interval $I$.

a. Show that each trajectory of this system lies on the surface of a (possibly degenerate) sphere centered at the origin $\left(0,0,0\right)$. [Hint: Compute $\frac{d}{dt}\left(x^2+y^2+z^2\right)$.] What does this say about the magnitude of the angular velocity vector?

b. Find all the critical points of the system, i.e. all points $\left(x_0,y_0,z_0\right)$ such that

$x\left(t\right)\equiv x_0$, $y\left(t\right)\equiv y_0$, $z\left(t\right)\equiv z_0$ is a solution. For such solutions, the angular velocity vector remains constant in the body system.

c. Show that the trajectories of the system lie along the intersection of a sphere and an elliptic cylinder of the form $y^2+2x^2=C$, for some constant $C$.[ Hint: Consider the expression for $dy/dx$ implied by Euler’s equations.]

d. Using the results of parts (b) and (c), argue that the trajectories of this system are closed curves. What does this say about the corresponding solutions?

e. Figure $5.19$ displays some typical trajectories for this system. Discuss the stability of the three critical points indicated on the positive axes.

Figure $5.19$Trajectories for Euler’s system