On action-angle variables (Q1261967)

The authors present a very brief, relatively elementary, and explicit construction of the familiar action-angle coordinates for completely integrable Hamiltonian systems. (That is, the existence of the symplectic structure \(\sum d\varphi^ \alpha \wedge dj^ \alpha\) on a neighbourhood \(T^ n \times U\) (\(U \subset \mathbb{R}^ n\)) of a compact preimage \(F_ q = f^{-1}(q)\), where \(f = (f_ 1,\dots,f_ n): M \to \mathbb{R}^ n\) is the mapping of a symplectic manifold \((M,\omega)\) determined by certain Poisson-commuting functions \(f_ 1,\dots,f_ n \in C^{\infty}(M)\).) Interesting historical comments, and 17 references are adjoined at the end.