Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom
The following main statement is proved, besides another, for the 2-torus \(T^ 2={\mathbb{R}}^ 2/2\pi {\mathbb{Z}}^ 2\). Theorem. Let L be an incompressible torus of class \(C^ 3\) imbedded in the hypersurface \(M=\{x\in T^*X:\) \(H(x)=h\), H being the Hamiltonian function\(\}\) (i.e. a torus whose imbedding into M induces a monomorphism of the fundamental groups \(\pi_ 1(L)\) to \(\pi_ 1(M))\) which is a Lagrangian manifold of \(T^*T^ 2\). If L has no closed orbits of the Hamiltonian flow then a natural projection \(\theta\) : \(T^*T^ 2\to T^ 2\) restricted to L is a diffeomorphism.
scientific article; zbMATH DE number 4105903