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 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 .
Without explicitly solving these equations, the motion can be described geometrically as follows:[1]
The motion is periodic, so traces out two closed curves, one on the ellipsoid, another on the plane.
If the rigid body is symmetric (has two equal moments of inertia), the vector describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor.