Research proof index

The angular momentum of a rigid body equals its inertia tensor applied to the angular velocity: L = I ω.

The first moment of the mass distribution about its own centre of mass vanishes: for nonzero mass, ρ of the centred j-th coordinate function is zero.

The rotational kinetic energy equals the mass integral of the local rotational speed squared: T = ½ ∫ |ω × r|² dm.

The center of mass of a solid sphere located at the origin is 0.

The angular velocity tensor is skew-symmetric, Ωᵀ = -Ω: it lies in the Lie algebra 𝔰𝔬(d). This is the litmus check that Ω = Ṙ Rᵀ is a genuine angular-velocity tensor, and follows by differentiating the orthogonality identity R Rᵀ = 1.

König's theorem in the body frame. The total kinetic energy M.kineticEnergy t, formed from the lab-frame point velocities, splits at the centre of mass (centerOfMass = 0) as T = ½ M ⟪V, V⟫ + rotationalKineticEnergy ω_body. The rotational energy is a frame-independent scalar, so it is evaluated here from the body-frame angular velocity ω_body...