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 (I2) 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 Poincaré map, even in the presence of resonant and chaotic orbits. This behavior allows us to define the calibrated angular momentum, the average angular momentum ($\overline{{L}_{z}}$) normalized by the amplitude of its fluctuation (${\sigma }_{{L}_{z}}$), as a numerical proxy for I2. It also implies that the amplitude of fluctuation in Lz, previously underappreciated, 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.