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

Gabern, Frederic and Jorba, Angel and Locatelli, Ugo (2005) On the construction of the Kolmogorov normal form for the Trojan asteroids. Nonlinearity, 18 (4). pp. 1705-1734. ISSN 0951-7715. doi:10.1088/0951-7715/18/4/017.

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In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them. The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.

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Additional Information:Copyright © Institute of Physics and IOP Publishing Limited 2005.b Recommended by A Chenciner. Received 18 October 2004, in final form 22 March 2005, Published 13 May 2005. Print publication: Issue 4 (July 2005). The authors thank A Giorgilli and J M Mondelo for letting them use, respectively, a software package for computer-algebra and a program for frequency analysis. P Robutel is also acknowledged for his comments on a previous version of this manuscript. FG and AJ have been supported by the MCyT/FEDER Grant BFM2003-07521-C02-01, the CIRIT grant 2001SGR-70 and DURSI. FG acknowledges the support of the Fulbright–GenCat postdoctoral programme. FG and UL have been supported by the research programme ‘Sistemi dinamici non lineari e applicazioni fisiche’, prot. no 2001018375 001, financed by MIUR. UL has been supported by the financing programme for young researchers of MURST (no 1707/98) and by the INdAM 2002/03 project ‘Sistemi dinamici interagenti’.
Issue or Number:4
Record Number:CaltechAUTHORS:GABnonlin05
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:900
Deposited By: Tony Diaz
Deposited On:03 Nov 2005
Last Modified:08 Nov 2021 19:05

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