| eBook ISBN: | 978-1-4704-4813-4 |
| Product Code: | MEMO/255/1218.E |
| List Price: | $78.00 |
| MAA Member Price: | $70.20 |
| AMS Member Price: | $46.80 |
| eBook ISBN: | 978-1-4704-4813-4 |
| Product Code: | MEMO/255/1218.E |
| List Price: | $78.00 |
| MAA Member Price: | $70.20 |
| AMS Member Price: | $46.80 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 255; 2018; 92 ppMSC: Primary 34; 70; 37
The author proves the existence of an almost full measure set of \((3n-2)\)-dimensional quasi-periodic motions in the planetary problem with \((1+n)\) masses, with eccentricities arbitrarily close to the Levi–Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V. I. Arnold in the 1960s, for the particular case of the planar three-body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group.
The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, a common tool of previous literature.
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Table of Contents
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Chapters
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1. Background and results
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2. Kepler maps and the Perihelia reduction
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3. The $\mathcal {P}$-map and the planetary problem
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4. Global Kolmogorov tori in the planetary problem
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5. Proofs
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A. Computing the domain of holomorphy
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B. Proof of Lemma 3.2
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C. Checking the non-degeneracy condition
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D. Some results from perturbation theory
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E. More on the geometrical structure of the $\mathcal {P}$-coordinates, compared to Deprit’s coordinates
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Additional Material
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The author proves the existence of an almost full measure set of \((3n-2)\)-dimensional quasi-periodic motions in the planetary problem with \((1+n)\) masses, with eccentricities arbitrarily close to the Levi–Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V. I. Arnold in the 1960s, for the particular case of the planar three-body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group.
The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, a common tool of previous literature.
-
Chapters
-
1. Background and results
-
2. Kepler maps and the Perihelia reduction
-
3. The $\mathcal {P}$-map and the planetary problem
-
4. Global Kolmogorov tori in the planetary problem
-
5. Proofs
-
A. Computing the domain of holomorphy
-
B. Proof of Lemma 3.2
-
C. Checking the non-degeneracy condition
-
D. Some results from perturbation theory
-
E. More on the geometrical structure of the $\mathcal {P}$-coordinates, compared to Deprit’s coordinates
