Precession

Suppose that some position vector ${\bf r}$ precesses (i.e., rotates) about the $z$-axis at the angular velocity ${\mit \Omega }$. If $x(t)$, $y(t)$, $z(t)$ are the Cartesian components of ${\bf r}$ at time $t$ then it follows from the analysis in the previous section that

$$\left(\begin{array}{c}x(t)\\ [0.5ex]y(t)\\ [0.5ex]z(t)\end{array}... ...right)\left(\begin{array}{c}x(0)\\ [0.5ex]y(0)\\ [0.5ex]z(0)\end{array}\right).$$ (A.93)

Hence, making use of the small angle approximations to the sine and cosine functions, we obtain

$$\left(\begin{array}{c}x(\delta t)\\ [0.5ex]y(\delta t)\\ [0.5ex]z... ...right)\left(\begin{array}{c}x(0)\\ [0.5ex]y(0)\\ [0.5ex]z(0)\end{array}\right),$$ (A.94)

which immediately implies that

$$\left(\begin{array}{c}\dot{x}\\ [0.5ex]\dot{y}\\ [0.5ex]\dot{z}\e... ...d{array}\right)\left(\begin{array}{c}x\\ [0.5ex]y\\ [0.5ex]z\end{array}\right),$$ (A.95)

or

$$\frac{d{\bf r}}{dt} = \mbox{\boldmath$\Omega$} \times{\bf r},$$ (A.96)

where $\Omega$$={\mit\Omega}\,{\bf e}_z$ is the angular velocity of precession. Because vector equations are coordinate independent, we deduce that the preceding expression is the general equation for the time evolution of a position vector ${\bf r}$ that precesses at the angular velocity $\Omega$.