3. Equilibrium and stability diagram
The equations of motion used are those defined by the Hamiltonian H, in a frame of reference rotating about the z -axis with the frequency , and the x -axis aligned with the long axis of the bar, that is,
where , are the canonical variables and the expression of H is
with the potential described in Sect. 2. Since is symmetric in z, it follows that and the equilibrium points lie in the plane . More specifically, the Lagrangian points on the x axis are obtained from the equation , and the Lagrangian points on the y -axis from .
Linearizing the equations of motion at and around a stationary point, we obtain the variational equations,
where the 's denote the linear variations and the potential second derivatives , , , and are constants evaluated at the stationary point. The terms , do not appear in Eq. (6) since vanishes at .
It is clear from Eq. (6) that the pair of equations for , separates from the rest and corresponds to an independent bounded harmonic motion if . In sufficiently thin disks is equivalent to requiring the positivity of mass at by Poisson's equation. Therefore we conclude that at each Lagrangian point a family of periodic orbits must start and continue transversally to the galaxy plane. Contrary to the axial orbits passing through the centre , these transverse orbits will not remain straight at non-infinitesimal amplitudes and their shape will be less obvious to determine.
Before computing these orbits it is useful to calculate their stability at low amplitude, which is just the stability of the relevant Lagrangian point. As determined in P90, one has to calculate the second derivatives of the potential , for a given pattern speed to find whether the motion around them is characterized by stability or any other type of instability.
In Fig. 2 we plot the position of the Lagrangian points in the relevant stability diagram discussed in P90 (see Fig. , p. 57). When increases from 0 to 0.8, the points remain simply unstable, moving toward the upper left of the figure; but the points see a transition from stability to complex instability (at ), moving downward across the transition curve. This is a crucial property of strong bars found in N -body simulations (P90).
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998