By Rudolf Dvorak, Sylvio Ferraz-Mello

Undesirable Hofgastein who made the very winning Salzburger Abend with indi- nous track from Salzburg attainable. targeted thank you additionally to the previous director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his beneficiant deepest donation. we should always now not omit our hosts Mr. and Mrs. Winkler and their staff from the lodge who made the remain particularly relaxing. None people will put out of your mind the final night, whilst the employees of kitchen below the le- ership of the prepare dinner himself got here to provide us as farewell the recognized Salzburger Nockerln, a conventional Austrian dessert. every body obtained loads of scienti?c enter through the lectures and the discussions and, to summarize, all of us had a spl- did week in Salzburg within the lodge Winkler. all of us desire to return back in 2008 to debate new effects and new views on a excessive point scienti?c typical 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 choice 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, email: zoran@aob. bg. a

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Namely, the region of stability is determined by evaluating IN up to the order N ¼ Nopt , where IN reaches its maximum value. Thus all the above described inequalities are fulﬁlled. 3. 1. THE MAPPING MODEL When seeking to determine a region of Nekhoroshev stability, a crucial factor is the choice of appropriate variables which should be such so as to ﬁt better the shape of the region that needs to be described. , the diﬀerence between the mean longitude of the asteroid and of Jupiter, and - is the longitude of the pericenter of the asteroid.

Math. Surv. 47, 57. : 1993, Cel. Mech. Dyn. 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.

The norm of the remainder (Equation (36)) is givenp inﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ terms of q ﬃ rather than IN . By using the coeﬃcient A an ‘eﬀective radius’ IN =ð1 À AÞ is deﬁned such that the size of the remainder can be expressed in terms of the level value IN rather than q. Precisely, for all IN such that qmax ðIN Þ < qÃ the following inequality holds for all qmin ðIN ÞOqOqmax ðIN Þ < qÃ ; ðNþ1Þ=2 IN jjRNþ1 jjq OBqÃ jjUNþ1 jj : 1ÀA ð43Þ Finally, consider a discrete time t ¼ 1; 2; . . ; corresponding to the number of iterations of the mapping (12).