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. Without explicitly solving these equations, Louis Poinsot was able to visualize the motion of the endpoint of the angular velocity vector. To this end he used the conservation of kinetic energy and angular momentum as constraints on the motion of the angular velocity vector . If the rigid rotor 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. Angular kinetic energy constraint 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. The path traced out on this ellipsoid by the angular velocity vector is called the polhode (coined by Poinsot from Greek roots for "pole path") and is generally circular or taco-shaped. Angular momentum constraint The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector is conserved in an inertial reference frame, so . The angular momentum vector can be expressed in terms of the moment of inertia tensor and the angular velocity vector which leads to the equation Since the dot product of and is constant, and itself is constant, the angular velocity vector has a constant component in the direction of the angular momentum vector . This imposes a second constraint on the vector ; in absolute space, it must lie on