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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

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

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

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

which immediately implies that

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

or

$\displaystyle \frac{d{\bf r}}{dt} =$   $\displaystyle \mbox{\boldmath$\Omega$}$$\displaystyle \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$ .


next up previous
Next: Curvilinear coordinates Up: Useful mathematics Previous: Rotational coordinate transformations
Richard Fitzpatrick 2016-03-31