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
Zero-velocity surface Zero-velocity surface The zero-velocity surface is a concept that relates to the N-body problem of gravity. It represents a surface a body of given energy cannot cross, since it would have zero velocity on the surface. It was first introduced by George William Hill. The zero-velocity surface is particularly significant when working with weak gravitational interactions among orbiting bodies. In the circular restricted three-body problem two heavy masses orbit each other at constant radial distance and angular velocity, and a particle of negligible mass is affected by their gravity. By shifting to a rotating coordinate system where the masses are