An Empirical Proxy for the Second Integral of Motion in Rotating Barred
  or Tri-axial Potentials
  We identify an effective proxy for the analytically-unknown second integral
of motion (I_2) for rotating barred or tri-axial potentials. Planar orbits of a
given energy follow a tight sequence in the space of the time-averaged angular
momentum and its amplitude of fluctuation. The sequence monotonically traces
the main orbital families in the Poincare map, even in the presence of resonant
and chaotic orbits. This behavior allows us to define the "Calibrated Angular
Momentum," the average angular momentum normalized by the amplitude of its
fluctuation, as a numerical proxy for I_2. It also implies that the amplitude
of fluctuation in L_z, previously under-appreciated, contains valuable
information. This new proxy allows one to classify orbital families easily and
accurately, even for real orbits in N-body simulations of barred galaxies. It
is a good diagnostic tool of dynamical systems, and may facilitate the
construction of equilibrium models.
