Rotational Motion of a Torque-Free Rigid Body

The animations on this page show the natural rotational motion of a free rigid body about its center of mass. The rotational motion of the body is depicted by showing the changing attitude of its kinetic energy ellipsoid. The principal geometrical axes of the ellipsoid (shown by white segments in all the animations) lie along the principal axes of inertia of the body, with the longest geometrical axis along the minor principal axis of inertia and the shortest along the major principal axis of inertia. The white segments in the animations thus show the instantaneous orientation of the principal axes frame of the body. The black lines depict the (nonrotating) inertial reference frame for comparison.

The animation on the right shows the rotational motion that results when the initial angular velocity of the body is close to its minor principal axis of inertia (also the longest axis of the ellipsoid). The blue circle shows the tip of the angular velocity vector. As the motion evolves, an inertial observer sees the angular velocity vector tracing out the herpolhode shown in green. However, an observer rotating with the body sees the same angular velocity vector tracing out the polhode shown in red. The polhode is closed, indicating that the body frame components of the angular velocity vector vary in a periodic fashion. The red dashed line shows the direction of the constant angular momentum vector. The pink curve shows the path traced out by a point on the minor principal axis of inertia.
   
The red polhode in the animation on the right appears to be rolling on the green herpolhode. This is not a mere coincidence. The seemingly complicated rotational motion of a free rigid body has a remarkably simple description that was first discovered by Poinsot: the rotational motion is such that the kinetic energy ellipsoid rolls without slipping on a fixed two-dimensional plane. The fixed plane, known as the invariant plane, is a stationary plane that is perpendicular to the angular momentum vector and tangent to the kinetic energy ellipsoid. The location and orientation of the invariant
plane are determined by the initial angular velocity and the initial orientation of the kinetic energy ellipsoid. The tip of the angular velocity vector is also the point of contact between the ellipsoid and the invariant plane. As the ellipsoid rolls without slipping, the point of contact traces out the herpolhode on the invariant plane and the polhode on the ellipsoid.

The animation on the left shows the kinetic energy ellipsoid rolling on the invariant plane in the case where the initial angular velocity is close to the major principal axis of inertia or the shortest geometrical axis of the ellipsoid. In this case the ellipsoid rolls on its flatter side tracing out the yellow herpolhode and the red polhode. The motion of the minor principal axis depicted by the pink curve consists of precession in which the axis goes around, and nutation in which the axis bobs up and down.
 
In both the animations above, the polhode is a closed curve either encircling the minor or the major axis, indicating that the angular velocity vector remains ''close'' to its initial value. This is because the states of spin about the minor and major axes are stable states for a rigid body. The last animation on this page shows the rotational motion that follows an  initial state of near spin about the intermediate principal axis of inertia. The red polhode almost connects the opposite ends of the intermediate axis, showing that the body transitions between spinning about the positive and negative intermediate axis. Even the pink curve showing the motion of the minor axis as well as the yellow herpolhode appear more erratic. This is because the state of spin about the intermediate axis is unstable, and even a small initial perturbation leads to a large departure.

The animations on this page depict the motion of an asymmetric rigid body whose principal moments of inertia are all unequal. The motion of an axisymmetric body having two of its principal moments of inertia equal is simpler to visualize.





                                                      The animations appearing on this page were created with the help of Debashish Bagg.