Numerical Simulation of rigid body rotations

The rotations of a general rigid body can be complicated: There is a feedback loop: the body’s instantaneous rotation vector is a function of the body’s moment of inertia (a rank 3 tensor) and the angular momentum (a vector); however, the moment of inertia in spatial coordinates is a function of the body’s orientation, which develops according to the body’s instantaneous rotation vector, which closes the feedback loop.

As a result, the rotation can be unstable, depending on a number of factors. These body gyrations are not intuitive because they are not observed outside friction-less free fall.

Here for example is a simulation of the rotation of a cuboid (gray) around its intermediary moment of inertia axis, which is unstable and flips the object around every couple of revolutions.

The two ellipsoids (transparent red and blue) are the Poinsot’s construction for the given body dimensions and angular momentum; their intersection curve has special meaning: there is a point (below visualized as a small white dot) that always exactly travels on it and that also remains on a plane fixed in space.(called the invariable plane). The body’s instantaneous rotation vector (shown in light blue) always passes through this point.