An infinitesimal Liouville-Arnold theorem as a criterion of reducibility for variational Hamiltonian equations (Q1208384)

Let a Hamiltonian vector field \(H_ f\) on a \(2n\)-dimensional analytic symplectic manifold \((M,\omega)\) be integrable on some invariant symplectic submanifold \({\mathcal T}\subset M\) of dimension \(2n\). Then \(\mathcal T\) is foliated into invariant tori \(T^ n_ p\) (\(p\) denotes a parameter), and the skew-normal bundle \((T{\mathcal T})^ \perp\subset T_{\mathcal T}M\) is invariant with respect to the flow \(S_ t\) of \(H_ f\). The author studies the problem if the flow \(S_ t\) on \((T{\mathcal T})^ \perp\) is reducible to a linear equation with constant coefficients along every torus \(T^ n_ p\). The result is applied to the stability problem after small Hamiltonian perturbations of lower order dimensional invariant tori. For the particular case \(N = n+1\), the result turns into a reducibility criterion for a zero-curvature equation.