Euler's Equations (rigid Body Dynamics) - Torque-free Solutions

Torque-free Solutions

For the RHSs equal to zero there are non-trivial solutions: torque-free precession. Notice that if I is constant (because the inertia tensor is the 3×3 identity matrix, because we work in the intrinsic frame, or because the torque is driving the rotation around the same axis so that I is not changing) then we may write

\mathbf{M} \ \stackrel{\mathrm{def}}{=}\ I \frac{d\omega}{dt}\mathbf{\hat{n}} = I \alpha \mathbf{\hat{n}}

where

α is called the angular acceleration (or rotational acceleration) about the rotation axis .

However, if I is not constant in the external reference frame (i.e. the body is moving and its inertia tensor is not the identity) then we cannot take the I outside the derivate. In this cases we will have torque-free precession, in such a way that I(t) and ω(t) change together so that their derivative is zero. This motion can be visualized by Poinsot's construction.

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