Hamilton's equations of motion from the SU(1,1)/U(1) symmetry of   Schwarzschild radial dynamics
  Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray. With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles. 
S. H. Castles