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 equals the angular momentum vector
Hence, the normal vector to the kinetic-energy ellipsoid at is proportional to, which is also true of the invariable plane. Since their normal vectors point in the same direction, these two surfaces will intersect tangentially.
Taken together, these results show that, in an absolute reference frame, the instantaneous angular velocity vector is the point of intersection between a fixed invariable plane and a kinetic-energy ellipsoid that is tangent to it and rolls around on it without slipping. This is Poinsot's construction.
Read more about this topic: Poinsot's Ellipsoid
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