- #1

aliens123

- 75

- 5

"Show that the components of the angular velocity along the space set of axes are given in terms of the Euler angles by

$$\omega_x = \dot{\theta} \cos \phi + \dot{\psi} \sin \theta \sin \phi,

\omega_y = \dot{\theta} \sin \phi - \dot{\psi} \sin \theta \cos \phi,

\omega_z = \dot{\psi} \cos \theta + \dot{\phi}$$"

I want to know what is wrong with the following line of reasoning:

A vector which has been rotated by the Euler angles can be given by

$$\vec{r'} = A \vec{r}$$ where $$A$$ is the matrix and has been given in the text. Then $$\Delta \vec{r} = \vec{r'} - \vec{r} = (A-I)\vec{r}.$$ So

$$\frac{d\vec{r}}{dt} = \frac{dA}{dt}\vec{r}.$$

We also know that a vector which has been rotated has the defining condition

$$\frac{d\vec{r}}{dt} = \vec{\omega} \times \vec{r}.$$

So by comparing $$\frac{dA}{dt}\vec{r}$$ to $$\vec{\omega} \times $$ we should be done. But this does not work. In fact, $$\frac{dA}{dt}\vec{r}$$ does not even have the same form as $$\vec{\omega} \times. $$