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
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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
The angular kinetic energy may be expressed in terms of the moment of inertia tensor
where
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
Angular momentum constraint
The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector
The angular momentum vector
which leads to the equation
Since the dot product of
The herpolhode is generally an open curve, which means that the rotation does not perfectly repeat, but the polhode is a closed curve.
Tangency condition and construction
These two constraints operate in different reference frames; the ellipsoidal constraint holds in the (rotating) principal axis frame, whereas the invariable plane constant operates in absolute space. To relate these constraints, we note that the gradient vector of the kinetic energy with respect to angular velocity vector
Hence, the normal vector to the kinetic-energy ellipsoid at
Taken together, these results show that, in an absolute reference frame, the instantaneous angular velocity vector
Derivation of the polhodes in the body frame
In the principal axis frame (which is rotating in absolute space), the angular momentum vector is not conserved even in the absence of applied torques, but varies as described by Euler's equations. However, in the absence of applied torques, the magnitude
where the
These conservation laws are equivalent to two constraints to the three-dimensional angular momentum vector
This construction differs from Poinsot's construction because it considers the angular momentum vector
Special Case
In the general case of rotation of an unsymmetric body, which has different values of the moment of inertia about the three principle axes, the rotational motion can be quite complex unless the body is rotating around the axis of maximum moment of inertia (stable rotation) or the axis of minimum moment of inertia (unstable rotation). The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principle axes. These cases include rotation of an prolate spheroid (the shape of an American football), or rotation of a oblate spheroid (the shape of a pancake). In this case, the angular velocity describes a cone, and the polhode is a circle. This analysis is applicable, for example, to the axial precession of the rotation of a planet (the case of an oblate spheroid.)
Applications
One of the applications of Poinsot's construction is in visualizing the rotation of a spacecraft in orbit.