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After studying action-angle variables and Eulers two-fixed-center problem in a course on mechanics and symplectic geometry, I understand that a two-fixed-center system is Liouville integrable and therefore (as long as the orbits are bounded) there exists a transformation to action-angle variables.

What I'm wondering is whether there is any book or paper where someone has done this transformation explicitly, or even just explored the question and found that the transformation can't be expressed in terms of elementary/known special functions. Is anybody here aware of such a source?

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  • $\begingroup$ Some time ago I wrote this post, maybe it can help. It treats the problem you are interested from the point of view of constructing a canonical transformation, in the form of a generating function, that trivializes the hamiltonain system. The method is via Hamilton-Jacobi equation and separation of variables: physics.stackexchange.com/questions/596627/… $\endgroup$
    – Futurologist
    Commented Mar 7 at 11:53

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