By Rudolf Dvorak, Sylvio Ferraz-Mello

Undesirable Hofgastein who made the very profitable Salzburger Abend with indi- nous tune from Salzburg attainable. distinctive thank you additionally to the previous director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his beneficiant inner most donation. we should always now not overlook our hosts Mr. and Mrs. Winkler and their staff from the lodge who made the remain particularly stress-free. None people will fail to remember the final night, while the employees of kitchen less than the le- ership of the prepare dinner himself got here to supply us as farewell the well-known Salzburger Nockerln, a conventional Austrian dessert. all people obtained loads of scienti?c enter in the course of the lectures and the discussions and, to summarize, all of us had a spl- did week in Salzburg within the inn Winkler. all of us wish to come back back in 2008 to debate new effects and new views on a excessive point scienti?c regular within the Gasteinertal. Rudolf Dvorak and Sylvio Ferraz-Mello Celestial Mechanics and Dynamical Astronomy (2005) 92:1-18 (c) Springer 2005 DOI 10. 1007/s10569-005-3314-7 FROM ASTROMETRY TO CELESTIAL MECHANICS: ORBIT decision WITH VERY brief ARCS (Heinrich ok. Eichhorn Memorial Lecture) 1 2 ? ANDREA MILANI and ZORAN KNEZEVIC 1 division of arithmetic, collage of Pisa, through Buonarroti 2, 56127 Pisa, Italy, electronic mail: milani@dm. unipi. it 2 Astronomical Observatory, Volgina 7, 11160 Belgrade seventy four, Serbia and Montenegro, electronic mail: zoran@aob. bg. a

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**Extra resources for A Comparison of the Dynamical Evolution of Planetary Systems: Proceedings of the Sixth Alexander Von Humboldt Colloquium on Celestial Mechanics Bad Hofgastein (Austria), 21-27 March 2004**

**Example text**

The correct series starts with the terms ð4Þ ¼ b2;2 U2;2 þ Á Á Á þ S0;m þ Á Á Á ; Ures ð24Þ where the value of the coeﬃcient b2;2 is chosen by the requirement that the numerator of (22) for p ¼ m þ 1; q ¼ 1 (or p ¼ 1; q ¼ m þ 1) is zero. Thus, the solution of Equation (21) for the terms S1;mþ1 is an arbitrary constant. This method eliminates the resonant terms of order m þ 2. FORMAL INTEGRALS AND NEKHOROSHEV STABILITY 37 In the same way it is possible to eliminate all the resonant terms of subsequent orders.

Astron. 57, 59. Namouni, F. and Murray, C. : 2000, Cel. Mech. Dyn. Astr. 76, 131. Nekhoroshev, N. : 1977, Russ. Math. Surv. 32(6), 1. : 1998, Non-linearity 11, 1465. , Gabern, F. : 2005, ‘The observed Trojans and the global dynamics around the Langvangian points of the Sun-Jupiter system’, Celest. Mech. Dynam. Astron. 92, 55–71. , E´rdi, B. : 2002, Cel.

Dvorak, R. : 1999, ‘Trojans in stable chaotic motion’ Celest. Mech. Dynam. Astron. 73, 117–126. : 1967, ‘Third-order stability of the long-period Trojan librations’, Astron. Journal 72, 10. , Gabern, F. : 2005, ‘The observed Trojans and the global dynamics around the Lagrangian points of the Sun-Jupiter system’, Celest. Mech. Dynam. Astron. 92, 55–71. : 2004, ‘On the stability of high inclined L4 and L5 Trojans’ Celest. Mech. Dynam. , 90, 139–148. : 2003, ‘Stability of Trojans with high inclined orbits’ In: F.