Inertietensoren

Inertietensoren, or inertia tensors, describe how mass is distributed within a rigid body with respect to a chosen origin. They encode the relationship between angular momentum and angular velocity: L = I · ω, where I is a 3×3 real symmetric matrix called the inertia tensor. For a continuous mass distribution with density ρ(r), the components in Cartesian coordinates are I_ij = ∫ (r^2 δ_ij − x_i x_j) ρ(r) dV, with r^2 = x^2 + y^2 + z^2. In a frame with the origin at the center of mass, the tensor generally has diagonal elements I_xx, I_yy, I_zz (the principal moments) and off-diagonal elements I_xy, I_xz, I_yz (products of inertia). The tensor depends on the chosen basis and transforms under a rotation R as I' = R I R^T.

The eigenvalues of I are the principal moments of inertia, and the corresponding eigenvectors define the principal axes. If the body rotates about a principal axis, angular momentum and angular velocity are parallel, and the rotation is described by a scalar moment of inertia. In general, L and ω are related by L = I · ω, and the dynamics follow τ = dL/dt, where τ is the external torque.

Common special cases include isotropic bodies, for which I is proportional to the identity, and planar bodies where rotation about a single axis dominates. The inertia tensor is central in rigid-body dynamics and is used to compute rotational kinetic energy T = 1/2 ω^T I ω and to solve Euler equations for rotating systems.