# Free Rotation of a Rigid Body: Poinsot Constructions

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The torque-free rotation of a rigid body can be described by Euler's three equations of motion: , and cyclic permutations*, *where , , are the principal moments of inertia and , , 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 and the kinetic energy . Euler's equations can be solved in closed form, giving , , in terms of Jacobi elliptic integrals.

Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: cuboid rigid body

Snapshot 2: cone construction for prolate rotor ()

Snapshot 3: Poinsot construction for oblate rotor ()

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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