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
Escape velocity velocity and the escape velocity. With a circular orbit, escape velocity is times the orbital speed. Thus the total escape velocity formula_29 when leaving one body orbiting a second and seeking to escape them both is, under simplified assumptions: where formula_31 for circular orbits. The last two columns will depend precisely where in orbit escape velocity is reached, as the orbits are not exactly circular (particularly Mercury and Pluto). Let "G" be the gravitational constant and let "M" be the mass of the earth (or other gravitating body) and "m" be the mass of the escaping body or projectile. At