A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems

In this paper we present an a-posteriori KAM theorem for the existence of an $(n-d)$-parameters family of $d$-dimensional isotropic invariant tori with Diophantine frequency vector $\omega\in \mathbb R^d$, of type $(\gamma,\tau)$, for $n$ degrees of freedom Hamiltonian systems with $(n-d)$ independent first integrals in involution. If the first integrals induce a Hamiltonian action of the $(n-d)$-dimensional torus, then we can produce $n$-dimensional Lagrangian tori with frequency vector of the form $(\omega,\omega_p)$, with $\omega_p\in\mathbb R^{n-d}$. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size $\rho$, and the corresponding error in the functional equation is $\varepsilon$. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if $\gamma^{-2} \rho^{-2\tau-1}\varepsilon$ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if $\gamma^{-4} \rho^{-4\tau}\varepsilon$ is small enough. The approach is suitable to perform computer assisted proofs.