Compactification of actions of \(\mathbb{R}^ k\) and Hamiltonian systems of toric type
Let \((M, \omega, H)\) be an Hamiltonian system endowed with a finite- dimensional space \(\mathcal A\) of first integral. The center of \(\mathcal A\) for the Poisson bracket defines an infinitesimal Hamiltonian action of \(\mathbb{R}^k\) on \(M\). Under the condition that \(\mathcal O\) is a compact orbit of this action, the author gives a necessary and sufficient condition for the system to admit a normal form of toric type in a neighborhood at \(\mathcal O\). This result generalizes the theorems of Arnol'd-Liouville, Eliasson, Nekhoroshev and its version with singularities. The key of the proof is a criterion of compactification for \(\mathbb{R}^k\)-actions which extends a result of J.-P. Dufour and P. Molino.
scientific article; zbMATH DE number 792827