Representation theory of the Galilean group to , and applying Hamilton's equations, we obtain the mass-velocity relation . The hypersurface is parametrized by this velocity In . Consider the stabilizer of a point on the orbit, (), where the velocity is . Because of transitivity, we know the unitary irrep contains a nontrivial linear subspace with these energy-momentum eigenvalues. (This subspace only exists in a rigged Hilbert space, because the momentum spectrum is continuous.) The subspace is spanned by , , and . We already know how the subspace of the irrep transforms under all operators but the angular momentum. Note that the rotation subgroup is
Angular velocity which shows a connection with the Lie group of rotations. We prove that angular velocity tensor is skew symmetric, i.e. formula_63 satisfies formula_64. A rotation matrix "A" is orthogonal, inverse to its transpose, so we have formula_65. For formula_66 a frame matrix, taking the time derivative of the equation gives: Applying the formula formula_68, Thus, "W" is the negative of its transpose, which implies it is skew symmetric. At any instant formula_70, the angular velocity tensor represents a linear map between the position vector formula_71 and the velocity vectors formula_72 of a point on a rigid body rotating around the