A Canonical Transformation to Eliminate Resonant Perturbations. I

• Published in: The Astronomical journal Vol. 162; no. 1; pp. 22 - 32
• Main Authors: Deme, Barnabás, Kocsis, Bence
• Format: Journal Article
• Language: English
• Published: Madison The American Astronomical Society 01.07.2021; IOP Publishing
• Subjects: ALGORITHMS; Astronomy; ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; CANONICAL TRANSFORMATIONS; Celestial mechanics; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DYNAMICAL SYSTEMS; HAMILTONIANS; HARMONIC OSCILLATORS; HARMONICS; Integrators; MECHANICS; Orbital mechanics; Orbital resonances (celestial mechanics); ORBITS; OSCILLATORS; Perturbation; Resonance; THREE-BODY PROBLEM; Transformations (mathematics)
• ISSN: 0004-6256; 1538-3881; 1538-3881
• DOI: 10.3847/1538-3881/abfb6d

We study dynamical systems that admit action-angle variables at leading order, which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical evolution may be integrated by orbit-averaging over the high-frequency angles, thereby evolving the orbit-averaged effect of the perturbations. It is well known that such integrators may be constructed via a canonical transformation, which eliminates the high-frequency variables from the orbit-averaged quantities. An example of this algorithm in celestial mechanics is the von Zeipel transformation. However, if the perturbations are inside or close to a resonance, i.e., the frequencies of the unperturbed system are commensurate; these canonical transformations are subject to divergences. We introduce a canonical transformation that eliminates the high-frequency phase variables in the Hamiltonian without encountering divergences. This leads to a well-behaved symplectic integrator. We demonstrate the algorithm through two examples: a resonantly perturbed harmonic oscillator and the gravitational three-body problem in mean motion resonance.