Bifurcation of a Whitney-smooth family of coisotropic invariant tori of a Hamiltonian system under small deformation of a symplectic structure (Q2784875)

The authors consider a perturbation problem for a completely integrable Hamiltonian system \((M,\omega_0^2,H_0)\), where \(M\) is even-dimensional manifold with symplectic structure \(\omega_0^2\), and \(H_0:M \mapsto \mathbb R\) is the system's Hamiltonian. The case is studied where both the Hamiltonian and the symplectic structure are perturbed: \(H_0 \mapsto H_0+\mu H_1\), \(\omega_0^2 \mapsto \omega_0^2+\mu \omega_1^2\). Here \(H_1: M \mapsto \mathbb R\) is a function, \(\omega_1^2\) is a \(2\)-form on \(M\), and \(\mu \) is a small parameter. The well-known Liouville-Arnold theorem asserts that in a neighborhood \(M'\) of invariant Lagrangian torus of the unperturbed system one can naturally define smooth free torus action preserving both \(H_0\) and \(\omega_0^2\). Let \(N\) be the corresponding quotient space, \(\pi :M' \mapsto N\) be the natural projection, and \(\overline{\omega}_1^2\) be the average of \(\omega_1^2\) with respect to the torus action. It turns out that the form \(\overline \omega_1^2\) determines on \(N\) a Poissonian structure. The authors suppose that the Poissonian system on \(N\) generated by the function \(H_0\) has a manifold \(E\) of stable equilibrium points. Under certain conditions they show that for sufficiently small \(\mu \) in a neighborhood of the manifold \(\pi ^{-1}(E)\) the perturbed Hamiltonian system has a massive Cantor set of coisotropic invariant minimal tori which is a part of a smooth family of coisotropic tori. This Cantor set can be separated in the phase space by means of a countable system of Diophantine inequalities involving smooth functions.