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Constructive Approaches to QP-Time-Dependent KAM Theory for Lagrangian Tori in Hamiltonian Systems

Renato Calleja, Alex Haro, Pedro Porras

TL;DR

This work proves an a-posteriori KAM theorem for Lagrangian tori in Hamiltonian systems with periodic or quasi-periodic time dependence by employing the parameterization method. It exploits a reduced, fibered symplectic framework to parameterize tori using $\theta$-dependent coordinates and a fixed external frequency vector $(\omega,\alpha)$, enabling a Quadratic convergence via a quasi-Newton iteration. The analysis provides explicit geometric and analytic estimates, including a novel, symmetric torsion expression that improves numerical efficiency and supports computer-assisted proofs. A isoenergetic variant and an extended geometric toolkit (including a geometrically adapted symplectic frame) broaden the applicability to non-perturbative regimes. The results offer a constructive, implementable pathway to compute invariant tori in time-dependent Hamiltonian systems, with potential applications in celestial mechanics and related dynamical systems.

Abstract

In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with $n$ degrees of freedom that depend periodically or quasi-periodically (QP) on time, with $\ell$ external frequencies. Such a system is described by a Hamiltonian function in the $2n$-dimensional phase space, $\mathscr{M}$, that depends also on $\ell$ angles, $\varphi\in \mathbb{T}^\ell$. We take advantage of the fibbered structure of the extended phase space $\mathscr{M} \times \mathbb{T}^\ell$. As a result of our approach, the parameterization of tori requires the last $\ell$ variables, to be precise $\varphi$, while the first $2n$ components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from $2(n+\ell)$ to $2n$. We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size $ρ_\infty$, provided suitable Diophantine $(γ,τ)$ conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, the details are given in the companion paper [CHP25]. This approach leverages precision and efficiency to compute invariant tori.

Constructive Approaches to QP-Time-Dependent KAM Theory for Lagrangian Tori in Hamiltonian Systems

TL;DR

This work proves an a-posteriori KAM theorem for Lagrangian tori in Hamiltonian systems with periodic or quasi-periodic time dependence by employing the parameterization method. It exploits a reduced, fibered symplectic framework to parameterize tori using -dependent coordinates and a fixed external frequency vector , enabling a Quadratic convergence via a quasi-Newton iteration. The analysis provides explicit geometric and analytic estimates, including a novel, symmetric torsion expression that improves numerical efficiency and supports computer-assisted proofs. A isoenergetic variant and an extended geometric toolkit (including a geometrically adapted symplectic frame) broaden the applicability to non-perturbative regimes. The results offer a constructive, implementable pathway to compute invariant tori in time-dependent Hamiltonian systems, with potential applications in celestial mechanics and related dynamical systems.

Abstract

In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with degrees of freedom that depend periodically or quasi-periodically (QP) on time, with external frequencies. Such a system is described by a Hamiltonian function in the -dimensional phase space, , that depends also on angles, . We take advantage of the fibbered structure of the extended phase space . As a result of our approach, the parameterization of tori requires the last variables, to be precise , while the first components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from to . We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size , provided suitable Diophantine conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, the details are given in the companion paper [CHP25]. This approach leverages precision and efficiency to compute invariant tori.

Paper Structure

This paper contains 29 sections, 11 theorems, 258 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and elementary constructions
  3. Basic notation
  4. QP Hamiltonian systems, symplectic bundles and invariant tori
  5. Linearized dynamics and reducibility
  6. Construction of a geometrically adapted frame
  7. A-posteriori KAM theory for quasi-periodic Hamiltonian systems
  8. KAM theorem for Lagrangian torus
  9. Some lemmas to control approximate geometric properties
  10. Estimates for cohomological equations
  11. Approximate lagrangianity
  12. Approximate symplectic frame
  13. Control of the torsion matrix
  14. Approximate reducibility
  15. Proof of the Lagrangian KAM theorem
  16. ...and 14 more sections

Key Result

Theorem 3.1

Let us consider an exact symplectic structure $\hbox{\boldmath $\omega$}= {\hbox{\rm d}} \hbox{\boldmath $\alpha$}$ and a Riemannian metric $\hbox{\boldmath $g$}$ on the open set ${\mathcal{M}}\subset{\mathbb R}^{2n}\times {\mathbb T}^{\ell}$. Let $h \in {{\mathcal{A}}}\left({\mathcal{M}}_\rho \tim Then, we assume that the following hypotheses hold. Under the above hypotheses, for each $0<\rho_\

Figures (1)

  • Figure 1: Two-dimensional torus expressed in cylindrical coordinates, each curve is a leaf of the torus fibration. The green leaf is the $K$ parameterization for a given $\varphi$, ${\mathcal{K}}_\varphi$.

Theorems & Definitions (31)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1: KAM theorem for quasi-periodic Hamiltonian systems
  • Remark 3.2
  • Lemma 4.1: Rüssmann estimates
  • Lemma 4.2: Approximate lagrangianity
  • proof
  • ...and 21 more