## 3. Equilibrium and stability diagramThe equations of motion used are those defined by the Hamiltonian
where , are the
canonical variables and the expression of with the potential described in
Sect. 2. Since is symmetric in 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. 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. [1],
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
© European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |