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
Sector mass spectrometer is simple. The physics are described by a single equation called the Lorentz force law. This equation is the fundamental equation of all mass spectrometric techniques and applies in non-linear, non-homogeneous cases too and is an important equation in the field of electrodynamics in general. where E is the electric field strength, B is the magnetic field induction, "q" is the charge of the particle, v is its current velocity (expressed as a vector), and × is the cross product. So the force on an ion in a linear homogeous electric field (an electric sector) is: in the direction of