Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem

Giorgilli, Antonio; Delshams, Amadeu; Fontich, Ernest; Galgani, Luigi; Simó, Carles

doi:10.1016/0022-0396(89)90161-7

Cited by 180 publications

(131 citation statements)

References 14 publications

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“…In our case this is, essentially, a pendulum depending on the parameters J 1 and J 2 . Using N (2) as an approximation, the xed points are located at cos 2 3 = n= q n 2 + p 2 sin 2 3 = p= q n 2 + p 2 J 3 = ( m q n 2 + p 2 )=(;2g)…”

Section: Reali Cationmentioning

confidence: 99%

Effective Stability for Periodically Perturbed Hamiltonian Systems

Jorba

Simó

1994

Hamiltonian Mechanics

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In this work we present a method to bound the di usion near an elliptic equilibrium point of a periodically time-dependent Hamiltonian system. The method is based on the computation of the normal form (up to a certain degree) of that Hamiltonian, in order to obtain an adequate number of (approximate) rst integrals of the motion. Then, bounding the variation of those integrals with respect to time provides estimates of the di usion of the motion.The example used to illustrate the method is the Elliptic Spatial Restricted Three Body Problem, in a neighbourhood of the points L 4 5 . The mass parameter and the eccentricity are the ones corresponding to the Sun-Jupiter case.

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Section: Reali Cationmentioning

confidence: 99%

Effective Stability for Periodically Perturbed Hamiltonian Systems

Jorba

Simó

1994

Hamiltonian Mechanics

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“…Indeed, it is proved below that for such energies the periodic orbit changes its stability several times as ε → 0, so again, the stability regions we find do not correspond to small oscillations that inherit their stability from the equilibrium state. For any finite n, for sufficiently small ε, γ(t) has a finite positive period and a(t) changes sign 7 as shown in Figure 3, so the behavior of the monodromy matrix A in the limit ε → 0 becomes non-trivial. Our main result is that there are wedges in the (µ, ε) space at which the eigenvalues of A are on the unit circle:…”

Section: Lemma 2 the Floquet Multipliers Of The T -Periodic Orbitmentioning

confidence: 99%

“…Nonetheless, by KAM theory, near such elliptic orbits there exists a positive measure set foliated by KAM-tori that corresponds to trajectories which remain forever near the elliptic trajectory. Furthermore, while other trajectories in this neighborhood may perhaps escape, this can take exponentially long time [18,7] (namely, such islands may correspond to highdimensional dynamical traps, generalizing the 2d stickiness phenomena). Thus, hereafter, an island in the multi-dimensional context will be defined as the small neighborhood of the elliptic orbit which is effectively stable [7], bearing in mind that only in the two degrees of freedom case this neighborhood is known to correspond to an invariant set.…”

Section: Introductionmentioning

confidence: 99%

Stability in High Dimensional Steep Repelling Potentials

Rapoport

Rom‐Kedar

Turaev

2008

Commun. Math. Phys.

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ABSTRACT. The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep is established for any finite n. Furthermore, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity properties of this limit system are destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established.

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“…These series do not converge in the usual sense but they do converge in suitable "Gevrey" spaces of formal series. Similar idea (to estimate the rate of divergence of formal series arising in normal forms) has been used in [9] to study the effective stability for a hamiltonian system in the vicinity of an elliptic equilibrium point. Our approach is different from those in [9], it is based on certain techniques which come from the calculus in Gevrey classes (see [5], [6], [13], [26]).…”

Section: J=2mentioning

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].…”

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confidence: 99%