The multi-scale KAM persistence without a scaling order for Hamiltonian systems
Weichao Qian, Yong Li, Xue Yang
TL;DR
This work addresses the persistence of multi-scale invariant tori in disordered Hamiltonian systems, motivated by celestial mechanics where scale ordering is not guaranteed. It introduces a readily verifiable non-degeneracy condition and develops a parameterized multi-scale KAM framework that builds near-identity symplectic changes to yield a Cantor family of real-analytic invariant tori on which the frequencies persist or converge to the unperturbed values as perturbations vanish. The core contributions include a concrete KAM step with truncation, extended small-divisor estimates, a solvable homological equation, and controlled frequency/coordinate transformations, plus an iterative scheme with convergence and precise measure estimates for the remaining parameter set. The paper also provides concrete examples demonstrating persistence in disordered multi-scale settings and clarifies how frequency components can be preserved or partially preserved on energy surfaces, broadening the applicability to realistic celestial systems and beyond.
Abstract
The persistence of invariant tori in multi-scale Hamiltonian systems is intrinsically linked to the stability of the N-body problem. However, the existing non-degeneracy conditions in disordered scenarios have been formulated too generally, making them difficult to apply directly to celestial mechanics. In this work, we present a readily verifiable non-degeneracy condition for the persistence of invariant tori in disordered multi-scale Hamiltonian systems.