principal axes

For any rigid body, no matter how irregularly shaped, there exist three mutually perpendicular axes, fixed in the body, called the principal axes of inertia. In the coordinate system defined by these axes, the inertia tensor is diagonal, with three values I_1, I_2, I_3 called the principal moments of inertia.

The significance: when a body rotates about one of its principal axes, the angular-momentum vector L is parallel to the angular-velocity vector ω, and the rotation is smooth and wobble-free. When a body rotates about any other axis, L and ω are not parallel — they point in different directions — and in the absence of external torque the body's spin axis traces a curve in the body frame. The result is a wobble, and it is responsible for the Chandler wobble of the Earth, the nutation of spinning tops, and the strange intermediate-axis instability of a tennis racket thrown spinning into the air.

For bodies with rotational symmetry (cylinders, spheres, cubes), the principal axes can be read off from the geometry — they coincide with the axes of symmetry. For asymmetric bodies they must be computed by diagonalising the inertia tensor. In every case, the three axes are orthogonal and the three principal moments are real — a consequence of the inertia tensor being symmetric and positive-definite. Rigid-body dynamics is much simpler when expressed in a principal-axis frame, and Euler's equations of motion are written most cleanly there.