Flow reconstruction and invariant tori in the spatial three-body problem

Palacián, Jesús F.; Sayas, Flora; Yanguas, Patricia

doi:10.1016/j.jde.2014.12.001

Cited by 7 publications

(7 citation statements)

References 45 publications

(118 reference statements)

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“…Taking τ > nL − 1 ensures that the size of all gaps within Γ γ τ,γ has small relative measure that, moreover, tends to 0 as γ → 0; here L is the highest derivative needed. In applications one can often do with L = 2, see, e. g., [19] for examples. In the case s = 1 of a single parameter, all derivatives up to L = n + r − 1 are needed in (2.7), leading to the condition…”

Section: Kam Theorymentioning

confidence: 99%

Persistence Properties of Normally Hyperbolic Tori

2018

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Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.

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Section: Kam Theorymentioning

confidence: 99%

Persistence Properties of Normally Hyperbolic Tori

2018

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“…Then, given a function (usually a polynomial) in terms of rectangular (or Moser) coordinates one applies the multivariate division algorithm with respect to the Gröbner basis, and the remainder of it yields this function in terms of the invariants alone. This procedure has been applied successfully in [41][42][43]46] in the setting of the three-body problem. Similarly, the generating functions appearing when pushing the normalization to higher orders can be simplified in the same way, taking into account that they are finite Fourier series in E whose coefficients are functions of the invariants a i,j .…”

Section: )mentioning

confidence: 99%

“…This occurs in many examples, such as the restricted three-body problem in the space [33,51]. For other three-body examples see for instance [41,42,46] where the Hamiltonian, after applying the Jacobi transformation to eliminate the translational symmetry, can be written as the sum of two Keplerian Hamiltonians plus a small perturbation dealing with the coupling terms of the two systems.…”

Section: Introductionmentioning

confidence: 99%

Normalization Through Invariants in n-dimensional Kepler Problems

2018

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We present a procedure for the normalization of perturbed Keplerian problems in n dimensions based on Moser 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.

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“…A study of quasi-periodic motions (including retrograde ones) bifurcating from relative equilibria appeared in Refs. 31,32 . However, the measure estimates obtained in those papers, based on a bit different framework (Withney regularity and no use of Birkhoff normal form; see, e.g., Ref.…”

Section: Introductionmentioning

confidence: 99%

“…However, the measure estimates obtained in those papers, based on a bit different framework (Withney regularity and no use of Birkhoff normal form; see, e.g., Ref. 32 (Theorem 5.1)), are not suited to the purposes of the paper, where the point of view is closer to Ref. 3 .…”

Section: Introductionmentioning

confidence: 99%

On the co-existence of maximal and whiskered tori in the planetary three-body problem

Pinzari

2018

Journal of Mathematical Physics

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In this paper we discuss about the possibility of coexistence of stable and unstable quasi-periodic kam tori in a region of phase space of the three-body problem. The argument of proof goes along kam theory and, especially, the production of two non smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for "properly-degenerate" systems, by a theorem of Nekhorossev and Miščenko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in [V.I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, 18, 85 (1963); p. 141]. PACS numbers: 02. Mathematical methods in physics a) gabriella.pinzari@math.unipd.it 1963's paper Ref. 3 -on the study of the dynamics of the so-called planetary problem, i.e.,the problem of (1 + N) point masses, one of which ("sun") is of "order one", while the remaining N ("planets") are of much smaller size, interacting through gravity. Indeed, when the reciprocal attraction among the planets, which is of much smaller order compared to the attraction between any planet and the sun, is set to zero, the planetary problem reduces to N uncoupled two-body problems ("unperturbed problem"). He considered the case of N planets in prograde configuration; i.e., revolving in the same verse, even though the question of the sense of rotation, at his time, was definitely of secondary importance, compared to the difficulties that had to be overcome and that we are going to recall. The lack of frequencies(a translationally invariant system with 1 + N bodies possesses 3N degrees of freedom. In the case of the planetary problem, it exhibits, as mentioned, only N < 3N frequencies) in the unperturbed problem was named by Arnold proper degeneracy. It represented a serious difficulty, if one wanted (as he was aiming to do) to apply Kolmogorov's theorem, Ref. 21 to the planetary problem.At a technical level, the appearance of the proper degeneracy consists, we might say, of a "loss of frequencies", caused by the "too many" (or, better Poisson non commuting 45 , see below) first integrals of motion. For such abundance, this kind of systems is often called super-integrable. Despite of the fact that the solutions of the two-body problem are known since Newton's times, a general, theoretical setting clearly explaining the phenomenon has been given only recently, thanks to the works by Nekhorossev and Miščenko and Fomenko, Refs. 27,30 (hereafter, nmf). The three authors proved a generalization of the best known Liouville-Arnold theorem, Ref.

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Flow reconstruction and invariant tori in the spatial three-body problem

Cited by 7 publications

References 45 publications

Persistence Properties of Normally Hyperbolic Tori

Persistence Properties of Normally Hyperbolic Tori

Normalization Through Invariants in n-dimensional Kepler Problems

On the co-existence of maximal and whiskered tori in the planetary three-body problem

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