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2013
Impact Factor

Patricia Yanguas

31006 Pamplona, Spain
Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra

Publications:

Meyer K. R., Palacián J. F., Yanguas P.
Normalization Through Invariants in $n$-dimensional Kepler Problems
2018, vol. 23, no. 4, pp.  389-417
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
Citation: Meyer K. R., Palacián J. F., Yanguas P.,  Normalization Through Invariants in $n$-dimensional Kepler Problems, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 389-417
DOI:10.1134/S1560354718040032
Meyer K. R., Palacián J. F., Yanguas P.
Stability of a Hamiltonian System in a Limiting Case
2012, vol. 17, no. 1, pp.  24-35
Abstract
We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the $1 : −1$ resonance case where the linearized system has double pure imaginary eigenvalues $\pm i \omega$, $\omega \ne 0$ and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.
This result implies the stability of the Lagrange equilateral triangle libration points, $\mathcal{L}_4$ and $\mathcal{L}_5$, in the planar circular restricted three-body problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh.
Keywords: stability, Lagrange equilateral triangle, KAM tori, Liapunov stable, planar circular restricted three-body problems, Routh’s critical mass ratio
Citation: Meyer K. R., Palacián J. F., Yanguas P.,  Stability of a Hamiltonian System in a Limiting Case, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 24-35
DOI:10.1134/S1560354712010030

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