Poinsots ellipsoid
lastmod 2020-09-20created 2020-09-203 edits
In classical mechanics, Poinsot's ellipsoid (Poinsot ellipsoid) is the set of angular velocities that a rigid body with a given rotational kinetic energy can have, and it forms an ellipsoid in angular velocity space. Applying the law of conservation of angular momentum to it gives the trajectory of the rigid body's motion. It is named after the French physicist Louis Poinsot.
Definition
Consider a rigid body with angular velocity \(\boldsymbol\omega\). Then the rotational kinetic energy \(T\) of the rigid body is (in the principal axes of inertia coordinate system)
\[ T=\frac12\omega_1^2I_1+\frac12\omega_2^2I_2+\frac12\omega_3^2I_3 \]
Here \(\boldsymbol\omega=(\omega_1,\omega_2,\omega_3)\) are the principal-axis components of the angular velocity, and
\[ \mathsf I=\begin{pmatrix} I_1\\ &I_2\\ &&I_3 \end{pmatrix} \]
are the principal-axis components of the rigid body's moment of inertia tensor. If the rigid body is not subject to external torque, its rotational kinetic energy is conserved. Therefore, the angular velocity of the rigid body is confined to an ellipsoid in angular velocity space with the given rotational kinetic energy \(T\). This is called Poinsot's ellipsoid. Because Poinsot's ellipsoid is defined along the principal axes of the rigid body, it rotates with the rigid body's rotation as if attached to the body.
(In the absence of external torque) the rigid body conserves not only energy but also angular momentum \(\mathbf L\).
\[ \mathbf L=\mathsf I\mathbf\omega \] .
Accordingly, the rotational kinetic energy can be written as follows.
\[ T=\frac12\mathbf\omega\cdot\mathbf L \] .
Therefore, the angular velocity of the rigid body is confined to a plane defined by the equation above, which is called the invariable plane (invariable plane), also known as the invariant plane. The invariable plane is a plane not in the principal axes but in the inertial frame of reference. That is, it is an absolute plane regardless of the rigid body's rotation.
The angular velocity of the rigid body therefore moves along the line of intersection of Poinsot's ellipsoid and the invariable plane, which makes it possible to compute the object's trajectory.