Inertietensor

Inertietensor, commonly referred to as the inertia tensor, is a mathematical object used in classical mechanics to describe how mass is distributed in a rigid body relative to its rotational motion. It is a second-order tensor that relates angular velocity to angular momentum through the relation L = I · ω, where L is the angular momentum, I is the inertia tensor, and ω is the angular velocity.

For a continuous rigid body with mass density ρ, the components of the inertia tensor are given by I_ij = ∫ ρ (r^2 δ_ij − x_i x_j) dV, where r is the position vector from a chosen origin, δ_ij is the Kronecker delta, and x_i are the coordinate components. In a discrete system of point masses, I_ij = ∑ m_k (r_k^2 δ_ij − x_i_k x_j_k). The tensor is symmetric (I_ij = I_ji), and its eigenvalues are the principal moments of inertia with corresponding eigenvectors defining the principal axes.

The inertia tensor is frame-dependent; it transforms under a change of coordinates as I' = R I R^T, where R is the rotation matrix. Its invariants include the trace, which equals the sum of the principal moments, and the determinant, related to their product.

Applications span rigid-body dynamics, spacecraft attitude control, biomechanics, and engineering analyses. In the body frame, Euler’s equations describe rotational motion using the inertia tensor: dL/dt + ω × L = τ, with L = I · ω. Special cases include isotropic bodies, for which I is proportional to the identity, and slender bodies, which yield characteristic principal moments.

The inertietensor is fundamental for predicting rotational behavior, stability, and response to applied torques, and it remains constant for rigid bodies with fixed mass distribution.