Properly-degenerate KAM theory   (following V. I. Arnold)

Chierchia, Luigi; Pinzari, Gabriella

doi:10.3934/dcdss.2010.3.545

Cited by 20 publications

(38 citation statements)

References 13 publications

(45 reference statements)

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“…The utility of Boigey-Deprit's coordinates became suddenly clear: switching (in order to overcome certain singularities of the chart) to a regularized version, called "RPS" coordinates, (acronym standing for "Regular, Planetary and Symplectic"), allowed them to derive the Birkhoff normal form of the planetary problem, to prove its non-degeneracy, and hence to complete the application of the Fundamental Theorem to the general problem. These results have been published in [6,7,9].…”

Section: Background and Resultssupporting

confidence: 58%

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Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem

Pinzari

2018

Memoirs of the AMS

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Chapter 1. Background and results Chapter 2. Kepler maps and the Perihelia reduction 2.1. The P-map vs rotations and reflections Chapter 3. The P-map and the planetary problem 3.1. A general property of Kepler maps 3.2. The case of the P-map Chapter 4. Global Kolmogorov tori in the planetary problem 4.1. A domain of holomorphy 4.2. A normal form for the planetary problem 4.3. A "multi-scale" KAM Theorem and proof of Theorem A Chapter 5. Proofs 5.1. Normalization of fast angles 5.2. Secular normalizations Appendix A. Computing the domain of holomorphy A.1. On the analyticity of the solution of Kepler equation A.2. Proof of Proposition 4.2 Appendix B. Proof of Lemma 3.2 Appendix C. Checking the non-degeneracy condition Appendix D. Some results from perturbation theory D.1. A multi-scale normal form theorem D.2. A slightly-perturbed integrable system Appendix E. More on the geometrical structure of the P-coordinates, compared to Deprit's coordinates Bibliography iii CHAPTER 1 1. BACKGROUND AND RESULTSFirstly, the P-map is well defined in the case of the planar problem, while JRBD coordinates are not. Everyone knows, in fact, that the starting point for the Radau-Jacobi reduction is the so-called "line of the nodes", the straight line determined by the intersection between the planes of the two orbits. When the orbits of the two planets belong to the same plane, this is not defined. A similar circumstance arises for Boigey-Deprit's coordinates, since their construction relies on certain straight lines in the space, which again lose their meaning in case of co-planarity.The proof of Arnold's theorem given in [27,9] is not affected by such singularity, since, as said, it relies on RPS coordinates, which, at the expense of one more degree of freedom, are well defined for co-planar motions -in that case they reduce to the classical Poincaré coordinates. It has its consequences when one wants to compare results for the fully reduced systems, in space or in the plane. The singularity of the chart does not allow one to state that motions in the spatial problem with minimum number of independent frequencies starting with very small inclinations stay close to the corresponding planar motions. Notwithstanding further studies appearing in [28], where this problem is partially overcome (via the construction of regular coordinates for coplanar motions defined locally), it would be nice, in principle, to handle a global system of action-angle coordinates which completely reduces rotations and is shared simultaneously by the planar and the spatial problem.Secondly, the P-map is well adapted to reflection symmetries of the problem, while JRBD coordinates are not, as discussed in [25,29].1 In the framework of the study of canonical coordinates for the planetary system, by "partial reduction", we mean a system of canonical coordinates where a couple of conjugated coordinates consists of integrals (e.g., functions of the three components of the total angular momentum). By "full reduction", we mean a partial reduction where also another...

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Section: Background and Resultssupporting

confidence: 58%

“…Theorem 4.7 generalizes [6, Proposition 3] in two respects. The first generalization concerns the consideration of m ≥ 2 scales (in [6] only the case m = 2 was treated).…”

Section: A "Multi-scale" Kam Theorem and Proof Of Theorem Amentioning

confidence: 99%

Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem

Pinzari

2018

Memoirs of the AMS

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“…Arnol'd used the case where d = 3, while Chierchia & Pinzari obtain Arnol'd's results with only d = 2 [10,11]. On the other hand, Chierchia & Pusateri prove the following theorem for d = 1 (see [19] for the C ∞ case): Theorem 6.1.…”

Section: Properly Degenerate Kam Theorymentioning

confidence: 99%

Invariant tori for multi-dimensional integrable hamiltonians coupled to a single thermostat

Butler

2021

Preprint

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This paper demonstrates sufficient conditions for the existence of KAM tori in a singly thermostated, integrable hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2-which include the Nosé thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of Legoll, Luskin & Moeckel, (Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity, 22 (2009Nonlinearity, 22 ( ), pp. 1673Nonlinearity, 22 ( -1694 to prove that a 'typical" singly thermostated, integrable, real-analytic hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion.

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“…is conserved and we choose a reference frame {k (1) , k (2) , k (3) } so that k (3) is parallel to C.…”

Section: The Planetary (1 + N)-body Problemmentioning

confidence: 99%