Trojan dynamics well approximated by a new Hamiltonian normal form

TL;DR

This paper introduces a novel Hamiltonian normal form for Trojan body dynamics in the PCRTBP, improving the approximation of stable regions and matching numerical simulations closely.

Contribution

It adapts modern normal form techniques with a new variable manipulation, providing an accurate integrable approximation for Trojan dynamics.

Findings

Good agreement between normal form level curves and numerical surfaces of section

Effective estimation of stable libration regions

Step-by-step algorithm enabling extensions to complex models

Abstract

We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricted Three-Body Problem (PCRTBP), by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel (1977) to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behavior of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non--normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models.