Action-angle coordinates at singularities for analytic integrable systems
Let \((M,\sigma)\) be a real analytic symplectic manifold of dimension \(2n\) and \(F:=(F_ 1,...,F_ n): M\to {\mathbb{R}}^ n\) a mapping with n real analytic, functionally independent functions \(F_ i: M\to {\mathbb{R}}\) which are in involution. Then we say that the mapping F defines a real analytic integrable system. This paper is concerned with a normal form of F at its singularities, which gives a generalization of a well-known theorem by Arnol'd and Jost concerning action-angle coordinates. Let \(X_{F_ i}\) denote the Hamiltonian vector field with Hamiltonian \(F_ i\). Suppose that \(rank(dF_ 1,...,dF_ n)=k\) at a point \(p_ 0\in M\). Then it is easily shown that a compact orbit of n vector fields \(X_{F_ 1},...,X_{F_ n}\) through \(p_ 0\) is a k-dimensional torus. Furthermore, it is shown that, under certain nondegeneracy condition, this torus belongs to a family of k-tori \(\{\Phi_ h\}\) which are invariant under the flows of the vector fields \(X_{F_ 1},...,X_{F_ n}\). Here \(h\) is a parameter running over a domain \(V\) of \({\mathbb{R}}^ k\). Under suitable conditions on the family \(\{\Phi_ h\}\), we can establish the existence of `generalized' action-angle coordinates in a neighbourhood of each torus \(\Phi_ h\) so that the vector fields \(X_{F_ i}\) can be solved explicitly in this coordinate system. The conditions to be imposed on \(\{\Phi_ h\}\) are natural from the viewpoint of the theory of dynamical systems. In particular, if \(\text{rank}(dF_ 1,...,F_ n)=n\) at \(p_ 0\), this result reduces to the theorem by Arnol'd and Jost. On the other hand, if \(\text{rank}(dF_ 1,...,dF_ n)=0\) at \(p_ 0\), it reduces to a result proved by the author [Comment. Math. Helv. 64, No.3, 412-461 (1989; Zbl 0686.58021)] which establishes the convergence of Birkhoff normal form near a non-resonant equilibrium point for (real) analytic integrable systems.
scientific article; zbMATH DE number 959
scientific article