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I have a problem with one of my study questions for an oral exam:

Euler’s equation of motion around the $z$ axis in two dimensions is $I_z\dot{\omega}_z = M_z$, whereas it in three dimensions is $I_z\dot{\omega}_z =-(I_y-I_x)\omega_x\omega_y+M_z$, assuming that the $xyz$ coordinate systems is aligned with the principal axis. Why does Euler’s equation of motion for axis $z$ contain the rotational velocities for axes $x$ and $y$?

How can one explain this physically? I mean I can derive Euler's equation of motion, but how can I illustrate that the angular velocities are changing in 3 dimensions?

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As explained on Wikipedia, the nice tensor form of the equations is $$ \mathbf{I} \cdot \dot{\boldsymbol\omega} + \boldsymbol\omega \times \left( \mathbf{I} \cdot \boldsymbol\omega \right) = \mathbf{M} $$ This reduces to your equations if one diagonalizes the tensor of the moment of inertia $I$ and labels the diagonal entries etc.

The three components are mixed with each other because quantities like $\vec\omega$ and $\vec M$ are really associated with rotations in space and rotations around the axis $x,y,z$ don't commute with each other – unlike translations. Translations commute with each other which is why the 3 components in $\vec F=m\vec a$ don't mix with each other.

For example, take the Earth, rotate it by 90 degrees around the $x$ axis, then 90 degrees around $y$ axis, then you rotate back by 90 degrees but first around $x$ axis again, so that you aren't undoing the $y$ rotation immediately, but then you undo the $y$ rotation, too. You don't get back where you have been: instead, you end up rotating the Earth around the $z$ axis. We say that rotations form the group $SO(3)$ which is non-abelian, $gh\neq hg$. The moment of force wants to rotate the rigid body around an axis but because it was already rotating around another axis given by $\vec \omega$ and the rotations don't commute with each other, the effect of the moment of force also influences the "remaining third" component.

A natural way to write the vectors $\vec \omega, \vec M$ is actually an "antisymmetric tensor" – they're "pseudovectors", not actual vectors. At any rate, when you correctly derive the equations, you should reproduce what Euler got.

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