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 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: 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, is the point of contact between the ellipsoid and the plane. The motion is periodic, so traces out two closed curves, one on the ellipsoid, another on the plane. The closed curve on the ellipsoid is the polhode. The closed curve on the plane is the herpolhode. 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. The law of conservation of energy implies that in the absence of energy dissipation or applied torques, the angular kinetic energy is conserved, so . The angular kinetic energy may be expressed in terms of the moment of inertia tensor and the angular velocity vector where are the components of the angular velocity vector , and the are the principal moments of inertia when both are in the body frame. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector ; in the principal axis frame, it must lie on the ellipsoid defined by the above equation, called the inertia ellipsoid.