# Evolution of Euler's angles in time

The general motion of a rigid body over time can be determined in the body frame by solving Euler's equations, selecting the principal axes for the body axes. Also, Euler's angles can be used to transform from the rotating body frame to an inertial space frame. But, in my physics mechanics textbooks and on this site, I have found no discussion of evaluating the motion over time in the space frame.

I did find the motion over time in the space frame addressed on page 10 of the following reference:http://dma.ing.uniroma1.it/users/lss_da/MATERIALE/Textbook.pdf. (This reference was provided by @JAlex in response to the question How do the inertia tensor varies when a rigid body rotates in space? on this exchange.)

Is the approach on page 10 of the reference the standard way to determine the motion in time in the space frame? Is the motion in time in the space frame important? If so, why is it not addressed in many standard physics mechanics textbooks that address the general motion of a rigid body?

Perhaps quaternions can be used as mentioned in a response by @John Alexiou to Euler's Angles and Uniquely Defining the Orientation of a Rigid Body on this exchange?

• May 6, 2022 at 19:30
• Related answer showing steps for a rigid body simulation. May 6, 2022 at 19:30
• Read this great article/refrence on how to do 3D rigid body dynamics. May 6, 2022 at 19:40

Euler Equation \begin{align*} &\mathbf{I}\,{\dot{\omega}}+\mathbf\omega\times \left(\mathbf{I}\,\mathbf\omega\right)=\mathbf\tau\tag 1 \end{align*}

and the kinematic equation

\begin{align*} &\mathbf{\dot{\phi}}=\mathbf{A}(\mathbf{\phi})\,\mathbf{\omega}\tag 2 \end{align*}

all vector components and the inertia tensor must be given either in body system or in inertial system

the inertia tensor in inertial system :

\begin{align*} &\mathbf{I}_I=\mathbf{R}\,\mathbf{I}_B\,\mathbf{R}^T \end{align*}

equation (2) in inertial system: with \begin{align*} & \left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]_B=\mathbf{R}^T\,\mathbf{\dot{R}}\quad\Rightarrow\quad \mathbf{\omega}_B=\mathbf{J}_R(\mathbf{\phi})\,\mathbf{\dot{\phi}}_B~\Rightarrow~ \mathbf{\dot{\phi}}_B=\underbrace{\mathbf{J}_R^{-1}}_{\mathbf{A}}\,\mathbf{\omega}_B \end{align*} \begin{align*} &\mathbf{\dot{\phi}}_I=\underbrace{\mathbf{R}\,\mathbf{A}\mathbf{R^T\,}}_{\mathbf{A}(\mathbf{\phi})}\mathbf{\omega}_I \end{align*}

the rotation matrix $$~\mathbf{R}~$$ can build up from three rotation matrices for example

\begin{align*} \mathbf{R}&=\mathbf{R}_x\,\mathbf{R}_y\,\mathbf{R}_z\\ &= \left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \phi_{{x}} \right) &-\sin \left( \phi_{{x}} \right) \\ 0&\sin \left( \phi_{{x}} \right) &\cos \left( \phi_{{x}} \right) \end {array} \right] \, \left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \phi_{{y}} \right) &-\sin \left( \phi_{{y}} \right) \\ 0&\sin \left( \phi_{{y}} \right) &\cos \left( \phi_{{y}} \right) \end {array} \right] \, \left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \phi_{{z}} \right) &-\sin \left( \phi_{{z}} \right) \\ 0&\sin \left( \phi_{{z}} \right) &\cos \left( \phi_{{z}} \right) \end {array} \right]\tag 3 \end{align*}

• $$~\mathbf I~$$ inertia tensor
• $$~\mathbf\omega~$$ angular velocity vector
• $$~\mathbf\tau~$$ external torque vector
• $$~\mathbf{\dot{\phi}}~$$ angle velocity vector
• $$~\mathbf R~$$ rotation matrix between B-system and I-system
• $$~I~$$ inertial system
• $$~B~$$ body system

with equation (3) \begin{align*} & \mathbf{\omega}_B=\mathbf{J}_R(\mathbf{\phi})\,\mathbf{\dot{\phi}}_B\quad, \mathbf{J}_R=\left[ \begin {array}{ccc} \cos \left( \phi_{{y}} \right) \cos \left( \phi_{{z}} \right) &\sin \left( \phi_{{z}} \right) &0 \\ -\cos \left( \phi_{{y}} \right) \sin \left( \phi_ {{z}} \right) &\cos \left( \phi_{{z}} \right) &0\\ \sin \left( \phi_{{y}} \right) &0&1\end {array} \right] \quad\Rightarrow\\ &\mathbf A_B=\left[ \begin {array}{ccc} {\frac {\cos \left( \phi_{{z}} \right) }{ \cos \left( \phi_{{y}} \right) }}&-{\frac {\sin \left( \phi_{{z}} \right) }{\cos \left( \phi_{{y}} \right) }}&0\\ \sin \left( \phi_{{z}} \right) &\cos \left( \phi_{{z}} \right) &0 \\ -{\frac {\sin \left( \phi_{{y}} \right) \cos \left( \phi_{{z}} \right) }{\cos \left( \phi_{{y}} \right) }}&{\frac {\sin \left( \phi_{{y}} \right) \sin \left( \phi_{{z}} \right) }{\cos \left( \phi_{{y}} \right) }}&1\end {array} \right]\quad, \mathbf{A}_I=\left[ \begin {array}{ccc} 1&{\frac {\sin \left( \phi_{{x}} \right) \sin \left( \phi_{{y}} \right) }{\cos \left( \phi_{{y}} \right) }}&-{ \frac {\cos \left( \phi_{{x}} \right) \sin \left( \phi_{{y}} \right) } {\cos \left( \phi_{{y}} \right) }}\\ 0&\cos \left( \phi_{{x}} \right) &\sin \left( \phi_{{x}} \right) \\ 0&-{\frac {\sin \left( \phi_{{x}} \right) }{\cos \left( \phi_{{y}} \right) }}&{\frac {\cos \left( \phi_{{x}} \right) } {\cos \left( \phi_{{y}} \right) }}\end {array} \right] \end{align*}