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- W4366734182 abstract "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 $omegain mathbb R^d$, 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 $omega_pinmathbb R^{n-d}$. 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 $gamma^{-2} rho^{-2tau-1}varepsilon$ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if $gamma^{-4} rho^{-4tau}varepsilon$ is small enough. The approach is suitable to perform computer assisted proofs." @default.
- W4366734182 created "2023-04-24" @default.
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- W4366734182 date "2023-04-20" @default.
- W4366734182 modified "2023-10-01" @default.
- W4366734182 title "A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems" @default.
- W4366734182 doi "https://doi.org/10.48550/arxiv.2304.10205" @default.
- W4366734182 hasPublicationYear "2023" @default.
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