Poinsot's ellipsoid

In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the kinetic energy of the body and the three components of the angular momentum, expressed with respect to an inertial laboratory frame. The angular velocity vector ${\displaystyle {\boldsymbol {\omega }}}$ of the rigid rotor is not constant, but satisfies Euler's equations. The conservation of kinetic energy and angular momentum provide two constraints on the motion of ${\displaystyle {\boldsymbol {\omega }}}$.

Without explicitly solving these equations, the motion ${\displaystyle {\boldsymbol {\omega }}}$ can be described geometrically as follows:

• The rigid body's motion is entirely determined by the motion of its inertia ellipsoid, which is rigidly fixed to the rigid body like a coordinate frame.
• Its inertia ellipsoid rolls, without slipping, on the invariable plane, with the center of the ellipsoid a constant height above the plane.
• At all times, ${\displaystyle {\boldsymbol {\omega }}}$ is the point of contact between the ellipsoid and the plane.

The motion is quasiperiodic. ${\displaystyle {\boldsymbol {\omega }}}$ traces out a closed curve on the ellipsoid, but a curve on the plane that is not necessarily a closed curve.

• The closed curve on the ellipsoid is the polhode.
• The curve on the plane is the herpolhode.

If the rigid body has two equal moments of inertia (a case called a symmetric top), the line segment from the origin to ${\displaystyle {\boldsymbol {\omega }}}$ sweeps out a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.