Abstract
The study of the Trojan problem (i.e. the motion in the vicinity of the equilateral Lagrangian points L 4 or L 5) has a long history in the literature. Starting from a representation of the Elliptic Restricted 3-Body Problem in terms of modified Delaunay variables, we propose a sequence of canonical transformations leading to a Hamiltonian decomposition in the three degrees of freedom (fast, synodic and secular). From such a decomposition, we introduce a model called the ‘basic Hamiltonian’ H b , corresponding to the part of the Hamiltonian independent of the secular angle. Averaging over the fast angle, the 〈H b 〉 turns to be an integrable Hamiltonian, yet depending on the value of the primary’s eccentricity e′. This allows to formally define action-angle variables for the synodic degree of freedom, even when e′ ≠ 0. In addition, we introduce a method for locating the position of secondary resonances between the synodic libration frequency and the fast frequency, based on the use of the normalized 〈H b 〉. We show that the combination of a suitable normalization scheme and the representation by the H b is efficient enough so as to allow to accurately locate secondary resonances as well as higher order resonances involving also the very slow secular frequencies.
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Notes
- 1.
We symbolize with arctan (a, b) the function \(\mathrm{tan}^{-1}(a/b): \mathbb{R}^{2} \rightarrow \mathbb{T}^{1}\), of two variables, that maps the value of the arctangent to the corresponding quadrant in the coordinate system with b as the abscissa and a as the ordinate.
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Acknowledgements
During this work, RIP was fully supported by the Astronet-II Marie Curie Initial Training Network (PITN-GA-2011-289240).
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Páez, R.I., Locatelli, U., Efthymiopoulos, C. (2016). The Trojan Problem from a Hamiltonian Perturbative Perspective. In: Gómez, G., Masdemont, J. (eds) Astrodynamics Network AstroNet-II. Astrophysics and Space Science Proceedings, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-23986-6_14
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DOI: https://doi.org/10.1007/978-3-319-23986-6_14
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