Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem

Yanguas, Patricia; Palacián, Jesús F.; Meyer, Kenneth R.; Dumas, H. Scott

doi:10.1137/070696453

Cited by 37 publications

(96 citation statements)

References 24 publications

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“…The analysis performed in Sections 3 and 4 is a continuation of the work initiated in [36] but the conclusions on the existence of the invariant 2-and 3-tori that we present in this paper are new. On the other hand part of the study in Section 5 has appeared in [9,21,22,32] but the analysis of these papers is incomplete.…”

Section: Introductionmentioning

confidence: 93%

“…The orbit space. In this subsection we summarize some general results from our earlier paper [36]. Let (M, Ω) be a symplectic manifold of dimension 2n, H 0 : M → R a smooth Hamiltonian which defines a Hamiltonian vector field Y 0 = (dH 0 ) # with symplectic flow φ t 0 .…”

Section: Perturbation Theoremsmentioning

confidence: 99%

“…The averaged Hamiltonian for the planar problem. Here we briefly summarize the normalization and reduction as given in [36]. For us the lunar problem is the restricted three-body problem where the infinitesimal is close to one of the primaries.…”

Section: Perturbation Theoremsmentioning

confidence: 99%

“…Some of the results of Section 2 are classic, some others are recent. Some of the results collected in this section appeared in [36] but Subsections 2.2, 2.3, and 2.4 are new (leaving apart Theorem 2.4). Section 3 contains the application to the planar lunar problem, leading to the existence of families of KAM 2-tori around the near circular orbits.…”

Section: Introductionmentioning

confidence: 99%

“…In an earlier paper [36] the authors investigated the existence, characteristic multipliers and stability of periodic solutions of a Hamiltonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. Our primary examples are the planar and spatial lunar problems of celestial mechanics, i.e., the restricted three-body problem where the infinitesimal particle is close to one of the primaries.…”

Section: Introductionmentioning

confidence: 99%

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Invariant tori in the lunar problem

Meyer¹,

Palacián²,

Yanguas³

2014

Publicacions Matemàtiques

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Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle flows. By averaging the perturbation over the bundle flow one obtains a Hamiltonian system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed system. Conditions on the critical points are given which insure KAM tori for the perturbed flow.These general theorems are used to show that the near circular periodic solutions of the planar lunar problem are orbitally stable and are surrounded by KAM 2-tori.For the spatial case it is shown that there are periodic solutions of two types, either near circular equatorial, that is, the infinitesimal particle moves close to the plane of the primaries following near circular trajectories and the other family where the infinitesimal particle moves along the axis perpendicular to the plane of the primaries following near rectilinear trajectories. We prove that the two solutions are elliptic and are surrounded by invariant 3-tori applying a recent theorem of Han, Li, and Yi.In the spatial case a second averaging is performed, and the corresponding orbit space (called the twice-reduced space) is constructed. The flow of the averaged Hamiltonian on it is given and several families of invariant 3-tori are determined using Han, Li, and Yi Theorem.

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Section: Introductionmentioning

confidence: 93%

Section: Perturbation Theoremsmentioning

confidence: 99%

Section: Perturbation Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Invariant tori in the lunar problem

Meyer¹,

Palacián²,

Yanguas³

2014

Publicacions Matemàtiques

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Applications

Kuehn

2014

Applied Mathematical Sciences

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Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

et al. 2018

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We study the dynamics of a family of perturbed three-degreesof-freedom (3-DOF) Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction of the two symmetries leads to a one-degrees-offreedom system that is completely analysed in the twice reduced space. All the relative equilibria, together with the stability and parametric bifurcations are determined. Moreover the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduce space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

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Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem

Cited by 37 publications

References 24 publications

Invariant tori in the lunar problem

Invariant tori in the lunar problem

Applications

Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

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