A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems

Benettin, Giancarlo; Galgani, Luigi; Giorgilli, Antonio

doi:10.1007/bf01230338

Cited by 163 publications

(98 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The authors made an attempt to relate the data for time dependence of the spectral entropy η(t), to the estimates of Refs. [95,96,97] for the rate of the Arnold diffusion. In 1964 Arnold has proved [98] that many-dimensional nonlinear systems are, in general, globally unstable due to a very peculiar diffusion (Arnold diffusion), for details see, e.g.…”

Section: Weak Chaosmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

307

292

View full text Add to dashboard Cite

A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Boseparticles are also considered.

show abstract

Section: Weak Chaosmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

307

292

View full text Add to dashboard Cite

show abstract

“…We can suppose that aj and f3j are G 2 functions. Moreover, taking {J^a + a) /2 instead of a we can arrange J^a = a still keeping (3.9), which implies /?i(^/\T]) = rj^{y^r)) with 7 in G' 2 We are going to show that (/) = (f) 1 is the desired diffeomorphism. We have to prove that (f) commutes with J\.…”

Section: Now Condition (24) Readsmentioning

confidence: 99%

“…Our aim is to show that aS w drj 2 A dy, Notice that da^ w 0 which implies n n da°^ A d77, -^db 0^ A dyj + dp/\drj^ w 0.…”

Section: Now Condition (24) Readsmentioning

confidence: 99%

“…Since then a number of new results about effective stability have appeared. Sharper estimates on the stability exponent a have been proved recently by Benettin, Gallavotti, Galgani, Giorgilli and others in the convex case (HQ is convex) in order to investigate stability problems in Celestial and Statistical mechanics [2], [3], [9]. A new approach to the effective stability of convex Hamiltonians, based on an analysis near the worst resonances of the system, has been recently proposed by Lochak [17].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Nekhoroshev type estimates for billiard ball maps

Gramchev¹,

Popov²

1995

Annales De L’institut Fourier

View full text Add to dashboard Cite

“…Stability estimates in nonlinear Hamiltonian dynamical systems are obtained by applications of either the KAM theorem (Kolmogorov 1954, Arnold 1963a,b, Moser 1962, or the Nekhoroshev theorem (Nekhoroshev 1977, Benettin et al 1985, , Lochak 1992, Pöschel 1993. A connection between the two theorems is provided by the theorem of superexponential stability (Morbidelli and Giorgilli 1995a,b).…”

Section: Introductionmentioning

confidence: 99%

Nekhoroshev stability estimates for different models of the Trojan asteroids

Efthymiopoulos¹

2004

Proc. IAU

View full text Add to dashboard Cite

Abstract.Estimates of the region of Nekhoroshev stability of Jupiter's Trojan asteroids are obtained by a direct (i.e. without use of the normal form) construction of formal integrals near the Lagrangian elliptic equilibrium points. Formal integrals are constructed in the Hamiltonian model of the planar circular restricted three body problem (PCRTBP), and in a mapping model (Sándor et al. 2002) of the same problem for small orbital eccentricities of the asteroids. The analytical estimates are based on the calculation of the size of the remainder of the formal series by a computer program. An analysis is made of the accumulation of small divisors in the series. The most important divisors introduce competing Fourier terms with sizes growing at similar rates as the order of truncation increases. This makes impossible to improve the estimates by considering nearly resonant forms of the formal integrals for particular near-resonances. Improved estimates were obtained in a mapping model of the PCRTBP. The main source of improvement is the use of better variables (Delaunay). Our best estimate represents a maximum libration amplitude D p = 10.60 . This is a quite realistic value which demonstrates the usefulness of Nekhoroshev theory.

show abstract

A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems

Cited by 163 publications

References 9 publications

The Fermi–Pasta–Ulam problem: Fifty years of progress

The Fermi–Pasta–Ulam problem: Fifty years of progress

Nekhoroshev type estimates for billiard ball maps

Nekhoroshev stability estimates for different models of the Trojan asteroids

Contact Info

Product

Resources

About