PC1672 Advanced dynamics

4.6 Euler's equations

If we use the principal axes as our coordinate axes, the components of the tensor $\underline{\underline{\bf I}}$ form a diagonal matrix

$\begin{displaymath}\underline{\underline{\bf I}}=\pmatrix{I_1&0&0\cr 0&I_2&0\cr 0&0&I_3\cr}\end{displaymath}$

and the action of the tensor takes the simple form,

$\begin{displaymath}\underline{\underline{\bf I}}\cdot\left(\omega_1{\bf e}_1^{\r... ...m P}+I_2\omega_2{\bf e}_2^{\rm P} +I_3\omega_3{\bf e}_3^{\rm P}\end{displaymath}$

This will allow us to simplify the equations of motion for a rotating body.

Torque is equal to rate of change of angular momentum in an inertial frame. In the body-fixed frame (a rotating frame) this becomes

$\begin{displaymath}\left({{\rm d}{\bf L}\over {\rm d}t}\right)_{\! r} +\hbox{\boldmath {$\omega$}}\times{\bf L}=\hbox{\boldmath {$\tau$}}\end{displaymath}$

Note that in this frame, the tensor $\underline{\underline{\bf I}}$ is constant, since it just depends on the positions of the atoms in the body and these do not move with respect to the body. Hence we can use ${\bf L}=\underline{\underline{\bf I}}\cdot\hbox{\boldmath {$\omega$ }}$ to write the equation of motion as

$\begin{displaymath}\underline{\underline{\bf I}}\cdot\dot{\hbox{\boldmath {$\ome... ...dot\hbox{\boldmath {$\omega$}}\right)=\hbox{\boldmath {$\tau$}}\end{displaymath}$

In terms of coordinate axes that lie along the principal axes of $\underline{\underline{\bf I}}$ , the components of the equation of motion are

$\displaystyle I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3$	$\textstyle =$	$\displaystyle \tau_1$
$\displaystyle I_2\dot\omega_2+(I_1-I_3)\omega_3\omega_1$	$\textstyle =$	$\displaystyle \tau_2$
$\displaystyle I_3\dot\omega_3+(I_2-I_1)\omega_1\omega_2$	$\textstyle =$	$\displaystyle \tau_3$

These are known as Euler's equations for rotational motion. They are particularly useful for studying systems where the torque lies in a fixed direction with respect to the body (or for free rotations where there is no torque). Note that they are symmetric under cyclic permutations of the indices 1, 2 and 3.

Textbook references

K&K 7.7 pages 320-322
F&C 9.3
B&O 7.6
M 12.8
M&T 11.8

Mike Birse
17th May 2000