Geodesic
Applicazioni del teorema di Nekhoroshev alla meccanica celeste
Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 71-95.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

The application of Nekhoroshev theory to selected physical systems, interesting for Celestial Mechanics, is here reviewed. Applications include the stability of motions in the weakly perturbed Euler-Poinsot rigid body and the stability of the so-called Lagrangian equilibria $L_4$ , $L_5$ in the spatial circular restricted three-body problem. The difficulties to be overcome, which require a nontrivial extension of the standard Nekhoroshev theorem, are the presence of singularities in the fiber structure the phase space, and the presence of «degenerate» variables (actions appearing in the perturbation, but not in the unperturbed system). 
@article{BUMI_2001_8_4B_1_a11,
     author = {Benettin, Giancarlo},
     title = {Applicazioni del teorema di {Nekhoroshev} alla meccanica celeste},
     journal = {Bollettino della Unione matematica italiana},
     pages = {71--95},
     publisher = {mathdoc},
     volume = {Ser. 8, 4B},
     number = {1},
     year = {2001},
     zbl = {1089.70008},
     mrnumber = {1438264},
     language = {it},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a11/}
}
TY  - JOUR
AU  - Benettin, Giancarlo
TI  - Applicazioni del teorema di Nekhoroshev alla meccanica celeste
JO  - Bollettino della Unione matematica italiana
PY  - 2001
SP  - 71
EP  - 95
VL  - 4B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a11/
LA  - it
ID  - BUMI_2001_8_4B_1_a11
ER  - 
%0 Journal Article
%A Benettin, Giancarlo
%T Applicazioni del teorema di Nekhoroshev alla meccanica celeste
%J Bollettino della Unione matematica italiana
%D 2001
%P 71-95
%V 4B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a11/
%G it
%F BUMI_2001_8_4B_1_a11
Benettin, Giancarlo. Applicazioni del teorema di Nekhoroshev alla meccanica celeste. Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 71-95. http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a11/

[A] H. Andoyer, Cours de Méchanique Celeste (Gautier-Villars, Paris 1923). | Jbk 49.0665.04

[BCG] G. Benettin-A. Carati-G. Gallavotti: A rigorous implementation of the Landau-Teller approximation for adiabatic invariants, Nonlinearity, 10 (1997), 479-505. | MR | Zbl

[BCF] G. Benettin-A. M. Cherubini-F. Fassò, Regular and chaotic behaviour of rigid bodies in fast rotation: a numerical study, In corso di stesura.

[BF1] G. Benettin-F. Fassò, Fast rotations of the symmetric rigid body: a general study by Hamiltonian perturbation theory, Part I, Nonlinearity, 9 (1996), 137-186. | MR | Zbl

[BF2] G. Benettin-F. Fassò, Classical «freezing» of plane rotations: a proof of the Boltzmann-Jeans Conjecture, Journ. Stat. Phys., 63 (1991), 737. | MR

[BG] G. Benettin-G. Gallavotti, Stability of Motions near Resonances in Quasi Integrable Hamiltonian Systems, Journ. Stat. Phys., 44 (1986), 293. | MR | Zbl

[BFG1] G. Benettin-F. Fassò-M. Guzzo, Fast rotations of the symmetric rigid body, a study by Hamiltonian perturbation theory. Part II, Gyroscopic rotations, Nonlinearity, 10 (1997), 1695-1717. | MR | Zbl

[BFG2] G. Benettin-F. Fassò-M. Guzzo, Nekhoroshev-stability of $L4$ and $L5$ in the spatial restricted three-body problem, Regular and Chaotic Dynamics, 3 (1998), 56-72. | MR | Zbl

[BFGG] G. Benettin-G. Ferrari-L. Galgani-A. Giorgilli, An Extension of the Poincaré-Fermi Theorem on the Non-Existence of Invariant Manifolds in Nearly-Integrable Hamiltonian Systems, Nuovo Cimento B, 72 (1982), 137-148. | MR

[BGG1] G. Benettin-L. Galgani-A. Giorgilli, A Proof of Nekhoroshev Theorem for Nearly-Integrable Hamiltonian Systems, Celestial Mechanics, 37 (1985), 1-25. | MR | Zbl

[BGG2] G. Benettin-L. Galgani-A. Giorgilli, Realization of Holonomic Constraints and Freezing of High Frequency Degrees of Freedom, in the Light of Classical Perturbation Theory. Part I, Comm. Math. Phys., 113 (1987), 87-103. | fulltext mini-dml | MR | Zbl

[BGG3] G. Benettin-L. Galgani-A. Giorgilli, Realization of Holonomic Constraints and Freezing of High-Frequency Degrees of Freedom in the Light of Classical Perturbation Theory. Part II, Comm. Math. Phys., 121 (1989), 557-601. | fulltext mini-dml | MR | Zbl

[D] A. Deprit, Free rotation of a rigid body studied in phase plane, Am. J. Phys., 55 (1967), 424.

[F] F. Fassò, The Euler-Poinsot top, a non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys., 47 (1996), 953-976. | MR | Zbl

[FGB] F. Fassò-M. Guzzo-G. Benettin, Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Comm. Math. Phys., 197 (1998), 347-360. | MR | Zbl

[FL] F. Fassò-Debra Lewis, Stability properties of the Riemann ellipsoids, preprint 2000. | fulltext mini-dml | MR | Zbl

[Ga] G. Gallavotti, Quasi-Integrable Mechanical Systems, in Critical phenomena, Random Systems, Gauge Theories, edito da K. Osterwalder and R. Stora, Les Houches, Session XLIII, 1984 (North-Holland, Amsterdam 1986). | MR | Zbl

[GDFGS] A. Giorgilli-A. Delshams-E. Fontich-L. Galgani-C. Simó, Effective Stability for a Hamiltonian System near an Elliptic Equilibrium Point, with an Application to the Restricted three Body Problem, J. Diff. Eq., 77 (1989), 167-198. | MR | Zbl

[GFB] M. Guzzo-F. Fassò-G. Benettin, On the stability of elliptic equilibria, Math. Phys. Electronic Journal, Vol. 4, No. 1 (1998). | MR | Zbl

[GG] Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celestial Mech., 37 (1985), 95-112. | MR

[Gi] A. Giorgilli, Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point, Ann. Inst. Henri Poincaré - Physique Thèorique, 48 (1988), 423-439. | fulltext mini-dml | MR | Zbl

[GM] M. Guzzo-A. Morbidelli, Construction of a Nekhoroshev like result for the asteroid belt dynamical system, Cel. Mech. & Dyn. Astr., 66 (1997), 255-292. | Zbl

[Gu] M. Guzzo, Nekhoroshev stability of quasi-integrable degenerate Hamiltonian Systems, Regular and Chaotic Dynamics, 4 (1999), 78-102. | MR | Zbl

[GM] M. Guzzo-A. Morbidelli, Construction of a Nekhoroshev like result for the asteroid belt dynamical system, Cel. Mech. & Dyn. Astr., 66 (1997), 255-292. | Zbl

[Lo] P. Lochak, Canonical perturbation theory via simultaneous approximation, Russ. Math. Surv., 47 (1992), 57-133. | MR | Zbl

[LN] P. Lochak-A. I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2 (1992), 495-499. | MR | Zbl

[MG] A. Morbidelli-M. Guzzo, The Nekhoroshev theorem and the Asteroid Belt dynamical system, Cel. Mech. & Dyn. Astr., 65 (1997), 107-136. | Zbl

[Ne1] N. N. Nekhoroshev, Behaviour of Hamiltonian systems close to integrability, Funct. Anal. Appl., 5 (1971), 338-339. (Funk. An. Ego Prilozheniya, 5 (1971), 82-83). | Zbl

[Ne2] N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Usp. Mat. Nauk, 32:6 (1977), 5-66 (Russ. Math. Surv., 32:6 (1977), 1-65). | MR | Zbl

[Ni] L. Niederman, Nonlinear stability around an elliptic equilibrium point in an Hamiltonian system, Nonlinearity, 11 (1998), 1465-1479. | MR | Zbl

[Po] H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste, Vol. 1 (Gautier-Villars, Paris, 1892). | Jbk 30.0834.08

[Pö] J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian Systems, Math. Z., 213 (1993), 187-216. | MR | Zbl