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 ω {displaystyle {oldsymbol {omega }}} 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 ω {displaystyle {oldsymbol {omega }}} . If the rigid rotor is symmetric (has two equal moments of inertia), the vector ω {displaystyle {oldsymbol {omega }}} describes a cone (and its endpoint a circle). This is the torque-free precession of the rotation axis of the rotor. 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 ω {displaystyle {oldsymbol {omega }}} 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 ω {displaystyle {oldsymbol {omega }}} . If the rigid rotor is symmetric (has two equal moments of inertia), the vector ω {displaystyle {oldsymbol {omega }}} 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 T   {displaystyle T } is conserved, so d T d t = 0 {displaystyle {frac {dT}{dt}}=0} . The angular kinetic energy may be expressed in terms of the moment of inertia tensor I {displaystyle mathbf {I} } and the angular velocity vector ω {displaystyle {oldsymbol {omega }}} where ω k   {displaystyle omega _{k} } are the components of the angular velocity vector ω {displaystyle {oldsymbol {omega }}} along the principal axes, and the I k   {displaystyle I_{k} } are the principal moments of inertia. Thus, the conservation of kinetic energy imposes a constraint on the three-dimensional angular velocity vector ω {displaystyle {oldsymbol {omega }}} ; in the principal axis frame, it must lie on an ellipsoid, called inertia ellipsoid. The ellipsoid axes values are the half of the principal moments of inertia. The path traced out on this ellipsoid by the angular velocity vector ω {displaystyle {oldsymbol {omega }}} is called the polhode (coined by Poinsot from Greek roots for 'pole path') and is generally circular or taco-shaped. The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector L {displaystyle mathbf {L} } is conserved in an inertial reference frame, so d L d t = 0 {displaystyle {frac {dmathbf {L} }{dt}}=0} . The angular momentum vector L {displaystyle mathbf {L} } can be expressedin terms of the moment of inertia tensor I {displaystyle mathbf {I} } and the angular velocity vector ω {displaystyle {oldsymbol {omega }}}