9. Andoyer Module¶
9.1. Introduction¶
The Andoyer module is used to represent the dynamical state of pair of planets in a standardized manner. Using a series of canonical transformations and re-scalings, the Hamiltonian governing the dynamics of a \(k\)-th order MMR, for \(k \le 3\), can (to leading order in eccentricity) be reduced the ‘Andoyer Hamiltonian’,
\[H_k(\Phi,\phi;\Phi') = 4\Phi^2 - 3\Phi'\Phi + \Phi^{k/2}\cos\phi\]
with the time measured in units of
\[\tau \propto \left(\mu\right)^{k} t~.\]
This form of the Hamltonian…