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
Cross product formula_123 is its velocity and formula_124 is the body's angular velocity. Since position formula_110 and velocity formula_123 are "true" vectors, the angular velocity formula_124 is a "pseudovector" or "axial vector". The cross product is used to describe the Lorentz force experienced by a moving electric charge formula_128: Since velocity formula_123, force formula_118 and electric field formula_132 are all "true" vectors, the magnetic field formula_133 is a "pseudovector". In vector calculus, the cross product is used to define the formula for the vector operator curl. The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in