Inertiamatriisin

Inertiamatriisin, often called the inertia matrix or inertia tensor, is a 3x3 symmetric tensor that encodes how mass is distributed with respect to a chosen origin and set of axes. It relates angular velocity to angular momentum through the relation L = I ω, where L is the angular momentum vector, ω is the angular velocity vector, and I is the inertia matrix. For a rigid body, the elements I_ij are defined by I_ij = ∑ m_k (r_k^2 δ_ij − x_i,k x_j,k) for a discrete system, or I = ∫ (|r|^2 I − r r^T) dm in the continuous form, with r the position vector from the origin.

The matrix depends on the chosen origin and orientation. In a body-fixed frame aligned with the principal axes, I is diagonal with the principal moments I1, I2, I3. Under a rotation R, the matrix transforms according to I' = R I R^T. Rotational dynamics are described by Euler’s equations: dL/dt + ω × L = τ, where τ is the applied torque. In the principal-axis frame, L = I ω componentwise, with L_i = I_i ω_i.

Common special cases arise from simple geometries. A uniform solid sphere has I = (2/5) m R^2 times the identity. A thin circular hoop has I along the axis of the hoop equal to m r^2, and about a diameter I = (1/2) m r^2.

Applications of the inertia matrix span spacecraft attitude control, robotics, biomechanics, and computer graphics, where it is used to predict rotational motion, assess stability, and design control strategies.