# The Euler-Lagrange equations for rigid body rotation [duplicate]

The equations of motion for rigid body rotation are:

$I\,\dot{\vec{\omega}}+\vec{\omega}\times I\,\vec{\omega}=\vec{\tau}$

How i can calculate this equations using Lagrangian method ?

If i use $L=\frac{1}{2}\vec{\omega}^T\,I\,\vec{\omega}$

i don't get the right equations.

Let $$S[ \omega, {\bf p},{\bf r}]= \int \left( \frac 12 I_1 \omega_1^2+ \frac 12 I_2 \omega_2^2+ \frac 12 I_3 \omega_3^2+{\bf p}\cdot (\dot{\bf r}+\omega \times{\bf r}) \right)$$ and vary all three vector variables ($\omega$, ${\bf p}$, and ${\bf r}$) to get an equation for each one. Then eliminate ${\bf p}$ and ${\bf r}$. You will end up with Euler's equations for the angular velocity $\omega$. Since this action is linear in the time derivative $\dot {\bf r}$ it is a Hamiltonian action principle. The vector ${\bf p}$ is Lagrange multiplier enforcing a Lin constraint. For more on Lin constraints see "Lin constraints, Clebsch potentials and variational principles" By Cendra and Marsden,  Physica D 27(1-2):63-89 (1987)
• Check my signs! I may have an error. We want to get $\dot {\bf L}+ \omega\times {\bf L}$=0 Sep 1, 2018 at 18:14