Abstract:
We present a procedure for the normalization of perturbed Keplerian problems in n dimensions based onMoser regularization of the Kepler problem and the invariants associated to the reduction process. The approach allows us not only to circumvent the problems introduced by certain classical variables used in the normalization of this kind of problems, but also to do both the normalization and reduction in one step. The technique is introduced for any dimensions and is illustrated for n=2,3 by relating Moser coordinates with Delaunay-like variables. The theory is applied to the spatial circular restricted three-body problem for the study of the existence of periodic and quasi-periodic solutions of rectilinear type.
Keywords:
Kepler Hamiltonian in n dimensions, perturbed Keplerian problems, Moser regularization, Delaunay and Delaunay-like coordinates, Keplerian invariants, regular reduction, periodic and quasi-periodic motions, KAM theory for properly degenerate Hamiltonians.
The authors have received partial support from Projects MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain, from MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of Spain and from the Charles Phelps Taft Foundation.
\Bibitem{MeyPalYan18}
\by Kenneth R. Meyer, Jes\'us F. Palaci\'an, Patricia Yanguas
\paper Normalization Through Invariants in $n$-dimensional Kepler Problems
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 4
\pages 389--417
\mathnet{http://mi.mathnet.ru/rcd330}
\crossref{https://doi.org/10.1134/S1560354718040032}
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https://www.mathnet.ru/eng/rcd330
https://www.mathnet.ru/eng/rcd/v23/i4/p389
This publication is cited in the following 4 articles:
Weichao Qian, Xue Yang, Yong Li, “Partial Frequency and Frequency Ratio in Multiscale KAM Formulism”, J Dyn Diff Equat, 2025
Jesús F. Palacián, Flora Sayas, Patricia Yanguas, “Invariant tori of rectilinear type in the spatial three-body problem”, Journal of Differential Equations, 399 (2024), 82
S. Ferrer, F. Crespo, J. L. Zapata, “Reduced 4D oscillators and orbital elements in keplerian systems: cushman-deprit coordinates”, Celest. Mech. Dyn. Astron., 132:11-12 (2020), 52
W. Qian, Y. Li, X. Yang, “Persistence of Lagrange invariant tori at tangent degeneracy”, J. Differ. Equ., 268:9 (2020), 5078–5112
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