Inertia tensors and principal axes

A point mass contributes to rotational inertia according to its distance perpendicular to the rotation axis. Mass lying on that axis contributes nothing to its moment, even when it lies far from the chosen origin. For many masses, add these contributions. The diagonal entries therefore contain sums of two squared coordinates; the off-diagonal entries contain negative products of different coordinates.

The tensor is symmetric because interchanging the coordinate labels leaves every mass contribution unchanged. It is positive semidefinite: rotational kinetic energy cannot be negative. An ideal collection confined to a line can have a zero moment about that line. A three-dimensional body with mass distributed away from every line through the origin instead has strictly positive principal moments.

A real symmetric matrix has an orthonormal eigenbasis. Choosing those directions as body axes makes the inertia tensor diagonal. The associated eigenvalues are the principal moments. This is a choice of coordinates attached to the object, not an operation that changes the object. Degenerate moments allow multiple equally valid principal directions within the degenerate eigenspace.

Along a principal direction, angular momentum is parallel to angular velocity. Along a general direction, multiplying different components by different moments usually changes the direction. A constrained rotor can maintain such a rotation only when its bearings provide the necessary torque. A free body can instead change its orientation and angular velocity while preserving its inertial angular momentum.

Shifting the origin changes inertia, even though changing only the orientation of axes does not change its eigenvalues. The parallel-axis theorem adds a tensor determined by the displacement from the center of mass. Never add a scalar shift independently to all diagonal entries without considering which direction the displacement points. The axis parallel to that displacement receives no added moment.

Use the vector lab to inspect one direction before doing tensor arithmetic. Predict that a unit vector at forty-five degrees has equal positive projections, then check it using the angle and magnitude controls. Multiplying those projections by unequal principal moments would tilt angular momentum toward the larger moment. The lab shows projections only; it neither multiplies a tensor nor simulates a rotating body.