# Understanding Euler's rotation theorem

According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear. - wikipedia

so can we say that the new axis of rotation is just the resultant of the previous too ?

Can you kindly explain this to me ? And show ( with some examples ) how can we mathematically predict this new axis of rotation ?

• kindly give some example of two "generalized rotations" happening simultaneously about two stationary axes- to get a picture of your problem! Mar 12, 2016 at 12:43
• For Euler's rotation theorem, see physics.stackexchange.com/q/19201/2451 and links therein. Mar 12, 2016 at 13:31
• Euler's rotation theorem applies only in odd dimensional spaces, and the notion of treating angular velocity as a vector is specific to three dimensional space. In four dimensional space, it is possible to have an object undergoing two simple rotations. Mar 12, 2016 at 14:40
• Sir, so in two-dimensional plane how can a object have two simple rotations ? Mar 12, 2016 at 15:16

Let us say you have a sequence of rotations about two axes, $\hat{x}$ and $\hat{y}$ by the angles $\varphi$ and $\theta$. You use rotation matrices to find the final orientation

$$\mathtt{E} = {\rm Rot}(\hat{x},\varphi)\,{\rm Rot}(\hat{y},\theta)$$

Now lets add some motion, and give the angles some speed

\begin{align} \dot{\varphi} & = \frac{{\rm d}\varphi}{{\rm d}t} \\ \dot{\theta} & = \frac{{\rm d}\theta}{{\rm d}t} \end{align}

The rotational velocity of this body is thus defined as

$$\vec{\omega} = \hat{x} \dot{\varphi} + {\rm Rot}(\hat{x},\varphi) \hat{y} \dot{\theta}$$

This is because $\hat{y}$ rotates about $\hat{x}$.

This motion is decomposed onto a rotation axis $\hat{z}$ and a rotation magnitude $\omega$

$$\omega = \| \hat{x} \dot{\varphi} + {\rm Rot}(\hat{x},\varphi) \hat{y} \dot{\theta} \| \\ \hat{z} = \frac{\vec{\omega}}{\omega}$$

Appendix I

To derive this you use the fact that the time derivative of a rotation matrix $\mathtt{E}$ is $$\dot{\mathtt{E}} = \vec{\omega} \times \mathtt{E}$$

Use the product rule to evaluate the left hand side (with $\mathtt{E} = \mathtt{R}_x \mathtt{R}_y$) as \begin{align}\dot{\mathtt{E}} &= \left( \frac{{\rm d}}{{\rm d}t} \mathtt{R}_x \right) \mathtt{R}_y + \mathtt{R}_x \left(\frac{{\rm d}}{{\rm d}t} \mathtt{R}_y \right)\\ &= ((\hat{x} \dot{\varphi}) \times \mathtt{R}_x ) \mathtt{R}_y +\mathtt{R}_x ( (\hat{y} \dot{\theta})\times \mathtt{R}_y) \\ &= (\hat{x} \dot{\varphi}) \times (\mathtt{R}_x \mathtt{R}_y) + (\mathtt{R}_x \hat{y} \dot{\theta})\times (\mathtt{R}_x \mathtt{R}_y) \\ &= (\hat{x} \dot{\varphi} + \mathtt{R}_x \hat{y} \dot{\theta}) \times (\mathtt{R}_x \mathtt{R}_y) \\ &= \vec{\omega} \times \mathtt{E} \,.\end{align}