On the construction of the Kolmogorov normal form for the Trojan asteroids

Gabern, Frederic; Jorba, Àngel; Locatelli, Ugo

doi:10.1088/0951-7715/18/4/017

Cited by 60 publications

(75 citation statements)

References 39 publications

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“…From the dynamical point of view, these trajectories are equivalent. These symmetries point out the fact that there are manifolds (even close to L 4 ) where the FM is degenerated (see Gabern et al 2005). These symmetries allow us to restrict the sample of initial conditions to the subset {( a , e ): a ≥ a 5 , e ≥ e 5 }.…”

Section: Global Structures Of Phase Spacementioning

confidence: 96%

“…On the other hand, analytical and semi‐analytical studies have provided important results that give insights on the stability problem of the Trojans. These works, mainly based on normal form computations, are generally developed using the restricted three‐body problem (RTBP) (Giorgilli et al 1989; Simó 1989; Giorgilli & Skokos 1997; Efthymiopoulos & Sándor 2005; Gabern, Jorba & Locatelli 2005). Recently, more sophisticated semi‐analytical models (Beaugé & Roig 2001; Gabern & Jorba 2004) have been used in order to study the stability near the Lagrangian points.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The resonant structure of Jupiter's Trojan asteroids - I. Long-term stability and diffusion

Robutel¹,

Gabern²

2006

Monthly Notices of the Royal Astronomical Society

111

100

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We study the global dynamics of the jovian Trojan asteroids by means of the frequency map analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the long‐term dynamics of the Trojans. Besides the secondary and secular resonances, that have already been found in other asteroid sets in mean motion resonance (e.g. main belt, Kuiper belt), we identify a new type of resonance that involves secular frequencies and the frequency of the great inequality, but not the libration frequency. Moreover, this new family of resonances plays an important role in the slow transport mechanism that drives Trojans from the inner stable region to eventual ejections. Finally, we relate this global view of the dynamics with the observed Trojans, identify the asteroids that are close to these resonances and study their long‐term behaviour.

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Section: Global Structures Of Phase Spacementioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

The resonant structure of Jupiter's Trojan asteroids - I. Long-term stability and diffusion

Robutel¹,

Gabern²

2006

Monthly Notices of the Royal Astronomical Society

111

100

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“…The algorithm used here is essentially the same as in [21], where we considered the more general case of the planetary problem with n + 1-bodies, and it is based on the procedure introduced in [11]. A detailed exposition including the adaptations that are convenient in the context of an explicit calculation using computer algebra may be found in [8]. We recall here the main steps for self-consistency purposes.…”

Section: Construction Of the Invariant Torusmentioning

confidence: 99%

Invariant tori in the Sun--Jupiter--Saturn system

Locatelli¹,

Giorgilli²

2007

Discrete &Amp; Continuous Dynamical Systems - B

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Abstract. We discuss the applicability of Kolmogorov's theorem on existence of invariant tori to the real Sun-Jupiter-Saturn system. Using computer algebra, we construct a Kolmogorov's normal form defined in a neighborhood of the actual orbit in the phase space, giving a sharp evidence of the convergence of the algorithm. If not a rigorous proof, we consider our calculation as a strong indication that Kolmogorov's theorem applies to the motion of the two biggest planets of our solar system. Introduction.We reconsider the problem of the applicability of KAM theory (see [13], [22] and [1]) to physical systems and, in particular, to planetary systems. As it is well known, the announcement of Kolmogorov's theorem has been immediately considered by the scientific community as a major step towards the solution of the classical problem of the stability of planetary motions. On the other hand, the results of several numerical calculations of the orbits seem to indicate that the applicability of the theorem to the whole solar system is an hopeless task. Indeed, Kolmogorov's theorem states that for most initial conditions the orbits of the system should be quasi-periodic with fixed frequencies, while the numerical calculations strongly indicate that the actual orbits exhibit a limited chaotic behavior which is superimposed to the quasi-periodic motion (see, e.g., [28], [15] and [23]). As a matter of fact, a rigorous check of the applicability of Kolmogorov's theorem at least to the main part of our solar system is still lacking.In this paper, we make a further step in this research field by considering the SunJupiter-Saturn (hereafter SJS) system in the framework of the general problem of three bodies. For this particular case, we produce a strong evidence of the existence of KAM tori, which are close to the actual orbits of the two biggest planets of our solar system.

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“…Contrastingly, our method is free from such a spurious diverging behavior and works even for such highly-excited molecules. Some other possible methods to improve validity ranges are using different styles of normalization [88] and using Kolmogorov normal form [89,90]. Both of the methods can be used combining with our method.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

Teramoto¹,

Toda²,

Komatsuzaki³

2014

Gregory S. Ezra

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Validity ranges of Lie Canonical Perturbation Theory (LCPT) are investigated in terms of non blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non blow-up regions enclose the region inside the separatrix of the Hamiltonian but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non blow-up regions do not necessarily cover the whole regions inside the HCN well.We propose a new perturbation method to improve non blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in [1,2].

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On the construction of the Kolmogorov normal form for the Trojan asteroids

Cited by 60 publications

References 39 publications

The resonant structure of Jupiter's Trojan asteroids - I. Long-term stability and diffusion

The resonant structure of Jupiter's Trojan asteroids - I. Long-term stability and diffusion

Invariant tori in the Sun--Jupiter--Saturn system

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

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