Research proof index

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...

The squared-speed integrand of kineticEnergy splits into the squared rotational speed |Ṙ (y − c)|², a term linear in the body-frame coordinate y − c, and the constant squared centre-of-mass speed ⟪V, V⟫.

The centre of mass of the moving mass distribution tracks the prescribed trajectory: for a body of nonzero mass, the centre of mass of massDistribution M t is exactly comTrajectory t. This is the decisive check that comTrajectory and orientation are wired correctly in RigidBodyMotion.

The velocity of a body point decomposes as v = Ṙ (y − c) + V: the rate of change of the orientation acting on the body-frame position, plus the centre-of-mass velocity.

A rigid body in pure translation (constant orientation) has every point moving with the centre-of-mass velocity: v = V.