The action-angle coordinates revisited: bi-Hamiltonian systems (Q1573832)

A method for using the action-angle variables to construct compatible bi-Hamiltonian structures for completely integrable systems is presented. The main theorem proved states that if the graph of a completely integrable Hamiltonian is a hypersurface of translation with respect to the action variables and each frequency depends only on a single action, then a master symmetry which transforms the system into a bi-Hamiltonian form is constructed. This result is applied to the Kepler problem and many invariant quantities are derived.