The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)

Nicolas Roy[1]

  • [1] Geometric Analysis Group, Institut für Mathematik, Humboldt Universität, Rudower Chaussee 25, Berlin D-12489, Germany.

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 2, page 383-397
  • ISSN: 0240-2963

Abstract

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Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions, we show the implication relations that exist between them, and we show the uniqueness of the fibration for non-degenerate Hamiltonians.

How to cite

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Roy, Nicolas. "The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)." Annales de la faculté des sciences de Toulouse Mathématiques 15.2 (2006): 383-397. <http://eudml.org/doc/10051>.

@article{Roy2006,
abstract = {Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions, we show the implication relations that exist between them, and we show the uniqueness of the fibration for non-degenerate Hamiltonians.},
affiliation = {Geometric Analysis Group, Institut für Mathematik, Humboldt Universität, Rudower Chaussee 25, Berlin D-12489, Germany.},
author = {Roy, Nicolas},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {KAM theorems; geometrical formulation; fibration of a symplectic manifold; Lagrangian tori; nondegenerate Hamiltonians},
language = {eng},
number = {2},
pages = {383-397},
publisher = {Université Paul Sabatier, Toulouse},
title = {The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)},
url = {http://eudml.org/doc/10051},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Roy, Nicolas
TI - The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 2
SP - 383
EP - 397
AB - Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions, we show the implication relations that exist between them, and we show the uniqueness of the fibration for non-degenerate Hamiltonians.
LA - eng
KW - KAM theorems; geometrical formulation; fibration of a symplectic manifold; Lagrangian tori; nondegenerate Hamiltonians
UR - http://eudml.org/doc/10051
ER -

References

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  2. A. D. Bruno, Local Methods in nonlinear differential equations, (1989), Springer-Verlag Zbl0674.34002MR993771
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  5. A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk. SSSR 98 (1954), 527-530 Zbl0056.31502MR68687
  6. J. Liouville, Note sur l’intégration des équations différentielles de la dynamique, J. Math. Pure App. 20 (1855), 137-138 
  7. H. Mineur, Réduction des systèmes mécaniques à n degrés de libertés admettant n intégrales premières uniformes en involution aux systèmes à variables séparées, J. Math. Pure Appl. 15 (1936), 221-267 Zbl0015.32401
  8. H. Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Number theory and dynamical systems (York, 1987) 134 (1989), 5-18, London Math. Soc., Cambridge Zbl0689.34039MR1043702
  9. A. Weinstein, Symplectic manifolds and their lagrangian subamnifolds, Adv. in Math. 6 (1971), 329-346 Zbl0213.48203MR286137
  10. A. Weinstein, Lagrangian submanifolds and hamiltonian systems, Ann. of Math. 98 (1973), 377-410 Zbl0271.58008MR331428

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