Free Rotation of a Rigid Body: Poinsot Constructions

The torque-free rotation of a rigid body can be described by Euler's three equations of motion: (Subscript[I, 1])(Subscript[(OverDot[ω]), 1])-((Subscript[I, 2])-(Subscript[I, 3]))(Subscript[ω, 2])(Subscript[ω, 3])=0, and cyclic permutations, where (Subscript[I, 1]), (Subscript[I, 2]), (Subscript[I, 3]) are the principal moments of inertia and (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 3]) are the angular velocities around their respective principal axes in the fixed-body coordinate system. There are two constants of the motion, the angular momentum (L^2)=((Subscript[I, 1])^2)((Subscript[ω, 1])^2)+((Subscript[I, 2])^2)((Subscript[ω, 2])^2)+((Subscript[I, 3])^2)((Subscript[ω, 3])^2) and the kinetic energy T=(1/2)((Subscript[I, 1])((Subscript[ω, 1])^2)+(Subscript[I, 2])((Subscript[ω, 2])^2)+(Subscript[I, 3])((Subscript[ω, 3])^2)). Euler's equations can be solved in closed form, giving (Subscript[ω, 1])(t), (Subscript[ω, 2])(t), (Subscript[ω, 3])(t) in terms of Jacobi elliptic integrals.

With no loss of generality, we will limit our considerations to a symmetric rotor with (Subscript[I, 2])=(Subscript[I, 1]). The rigid body will be assumed to be a cuboid of dimensions a×b×c (with b=a) but the same solutions apply to any rigid body with the same ellipsoid of inertia. The angular momentum is a constant vector L, oriented vertically. The instantaneous angular velocity ω, along with its component (Subscript[ω, 3]), is then found to precess around L with an angular velocity Ω t. This can be pictured as a red cone rolling around a stationary blue cone, shown in the "Poinsot cones" graphic.

Poinsot also proposed in 1834 a geometric construction that provides an elegant visual representation of rigid-body motion. The Poinsot ellipsoid, with principal axes  ((Subscript[I, 1])^(-1/2)),((Subscript[I, 2])^(-1/2)),((Subscript[I, 3])^(-1/2)), rolls on the invariable plane that is perpendicular to the constant angular-momentum vector. The path of the angular velocity vector ω within the ellipsoid is called the polhode. Its path on the invariable plane is called the herpolhode. For  symmetric rotors, both are circles. In the more general case, they have more complicated shapes. Goldstein summarizes the Poinsot construction in the Jabberwockian-sounding statement: the polhode rolls without slipping on the herpolhode lying in the invariable plane.

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Snapshot 1: cuboid rigid body

Snapshot 2: cone construction for prolate rotor (a=b<c)

Snapshot 3: Poinsot construction for oblate rotor (a=b>c)

References:

H. Goldstein, Classical Mechanics, Cambridge, MA: Addison–Wesley, 1950 pp. 156–163.

J. L. Synge and B. A. Griffiths, Principles of Mechanics, 2nd ed., New York: McGraw–Hill, 1949 pp. 418–429.

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Contributed by: S. M. Blinder

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