I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of that Jacobian. The equation is:
$ \ddot a-2\dot b -a= \displaystyle\frac{\partial U}{\partial a} $
$ \ddot b +2 \dot a-b = \displaystyle\frac{\partial U}{\partial b} $
with theU defined by $U=(1- \mu)/ \ r_{1} + \mu /r_{2} $ and
$ r_{1} = \sqrt {(a+\mu)^2 +b²} $ and $ r_{2} = \sqrt{(a-1+\mu)+b²} $.
The linearization need to be made in the equilibrium positions of the system (obviously).
I have obtained the five equlibrium positions explicitily, these are the five Lagrangian points, call it $ (a_{i}, b_{i}) $ for i=1,2,3,4,5, and the condition is that, $\mu=1/2 $ i.e., that the two principal masses are equal (interpreting the problem as the restricted three body problem).
How can now, linearize the equation aroun this five points, and calculate the eigenvalues of the resulting Jacobian? I know how is the linearization for ordinary systems, but, how can that work for this partial system?