# 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.