Moment of Inertia Tensor

The kinetic energy of a rigid body rotating with angular velocity ω is T = (1/2) ωᵀ I ω, and angular momentum is L = I ω. For these expressions to give the right answer for any direction of ω, I must transform like a tensor — a 3×3 matrix with I_ij = ∫ ρ(δ_ij r² − x_i x_j) dV. Diagonal entries I_xx, I_yy, I_zz are the familiar moments about each axis; off-diagonals (products of inertia) like I_xy = −∫ρ x y dV measure how rotation about x couples angular momentum into y. Only when those are zero (an axis of symmetry) does I reduce to a single scalar number.

Because I is real and symmetric, it has three orthogonal eigenvectors and three real eigenvalues. The eigenvectors are the principal axes; the eigenvalues are the principal moments I₁, I₂, I₃. Rotated into this body-fixed basis, I is diagonal — products of inertia vanish. Spinning about a principal axis: L is parallel to ω, so the body rotates steadily without needing any torque (in free space). For a uniform brick the principal axes are simply the three symmetry axes; for an asymmetric body you find them by diagonalizing I.

Parallel axis theorem: I_axis = I_cm + Md², where d is the perpendicular distance from the axis to a parallel axis through the center of mass. The full tensor version is I_O = I_cm + M (d² 1 − d ⊗ d), where d is the displacement vector from the CM to the new origin O. Used to compute moments about points other than the CM — physical pendulum periods, rotor balancing, robot arm dynamics. The theorem holds because the cross-term integral ∫ r · d dm vanishes when r is measured from the CM.

Toss a tennis racket spinning about its handle (smallest I) or about the axis perpendicular to the face (largest I) and it spins stably. Toss it about the middle axis (across the face, in the racket's plane) and after a fraction of a turn it flips end-over-end, then flips back, and so on. Cosmonaut Vladimir Dzhanibekov filmed this for a wing-nut on Soyuz T-13 in 1985. The Euler equations show that rotations about extremal-I axes are stable to perturbation; rotations about the intermediate-I axis are linearly unstable. Critical for picking which axis a spinning satellite should sleep around.

Spin-stabilized satellites must rotate about their major (largest I) axis to stay stable; with internal damping, energy bleeds away while angular momentum is conserved, so rotation drifts toward the configuration of minimum kinetic energy at fixed L — the major axis. Explorer 1 (1958) was designed to spin about its long, thin axis (minimum I) and embarrassingly tumbled into rotation about the perpendicular axis instead. Modern 3-axis stabilized spacecraft use reaction wheels and momentum bias to actively pick a stable orientation.

Plot the surface { x : xᵀ I x = 1 } in body-fixed coordinates. This is an ellipsoid whose principal semi-axes are 1/√I₁, 1/√I₂, 1/√I₃ along the principal directions. Poinsot's construction: in torque-free motion the angular velocity vector ω rolls (without slipping) on a fixed plane perpendicular to L, and traces a curve called the polhode on the ellipsoid. The polhode reveals stability geometrically — closed curves around the major and minor axes (stable), figure-eight separatrices through the intermediate axis (unstable Dzhanibekov flips).