A semi-numerical perturbation method for separable hamiltonian systems

Abstract

A detailed account is given of a semi-numerical perturbation method which has been proposed and improved upon in a succession of previous papers. The method analyses the first order effect of a small perturbation applied to a non-trivial two-degreeof-freedom separable Hamiltonian system (including the description of resonances) and construct approximate surfaces of section of the perturbed system. When the separable Hamiltonian system is already the description of a resonance, as it is the case in the problems we have investigated in the previous papers, these resonances are actually secondary resonances. The key role of the method is the numerical description of the angle-action variables of the separable system. The method is thus able to describe the perturbations of non-trivial separable systems and is not confined to the analysis of a small neigborhood of their periodic orbits.

Approximate integrals of motion

“…It implies the availability of a close and integrable Hamiltonian whose tori can be simply mapped to the target tori; it also implies mappings from different toy tori for different orbit families (Kaasalainen & Binney 1994). Moreover, there are general methods of semi-numerical perturbation that take into account the full 'distortion' of the invariant tori (Henrard 1990).…”

1:1 Ground-track resonance in a uniformly rotating 4th degree and order gravitational field

“…A formal way to deal with this system is to treat H 2d as a small perturbation to the integrable system H 2dof = H 0 (Henrard 1990). However, in our study, the perturbation from H 2d is not necessarily to be small values, due to the large variations of e and i.…”

Secular Orbital Evolution, Proper Elements, and Proper Frequencies for Near-Earth Asteroids: A Comparison between Semianalytic Theory and Numerical Integrations

“…The other is a semianalytic approach due to Lemaître and Morbidelli (1994), which is more appropriate for the orbits with either large eccentricities or large inclinations; it is based on the classical averaging method (Arnold 1976) in its revisited version (Henrard 1990). A similar method had already been used by Williams to obtain a set of proper elements that led to the understanding of the secular resonances resulting from two secular frequencies being equal (Williams 1969, 1979, Williams and Faulkner 1981.…”