By Rudolf Dvorak, Sylvio Ferraz-Mello
Undesirable Hofgastein who made the very profitable Salzburger Abend with indi- nous song from Salzburg attainable. certain thank you additionally to the previous director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his beneficiant deepest donation. we must always now not omit our hosts Mr. and Mrs. Winkler and their staff from the lodge who made the remain fairly relaxing. None people will fail to remember the final night, whilst the employees of kitchen below the le- ership of the cook dinner himself got here to supply us as farewell the recognized Salzburger Nockerln, a standard Austrian dessert. every body acquired loads of scienti?c enter throughout 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 commonplace 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 selection 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, e 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
Dvorak, R. : (2000), Cel. Mech. Dyn. Astr. 78, 125. , Giorgilli, A. : 2004, J. Phys. A 37, 10831–10858. : 1988, Cel. Mech. 43, 303. : 1997, Cel. Mech. Dyn. Astr. 65, 149. , Guzzo, M. : 1998, Commun. Math. Phys. 197, 347. : 1988, Ann. Inst. H. Poincare´ 48, 423. : 1999, In: C. ), Hamiltonian Systems with Three or More Degrees of Freedom, Kluwer, Dordrecht. Giorgilli, A. : 1997, Astron. Astrophys. 317, 254. Hadjidemetriou, J. : 1991, In: A. E. ), Predictability, Stability and Chaos in N-Body Dynamical Systems, Plenum Press, New York, pp.
In practice, even without such rigorous bounds, the results are in general quite precise, because the roundoﬀ error does not have a serious inﬂuence on the sizes of the dominant terms in the series, which are determined mainly by the mechanism of accumulation of small divisors. An alternative way to produce formal integrals with exponentially small remainders is the direct method due to Whittaker (1916), Cherry (1924) and Contopoulos (1960). , for which the point p ¼ 0 is an elliptic equilibrium point.
1 is equal to zero, which is obviously wrong. This is due to the fact that the size of the remainder during the whole excursion from IN to I0N ! 1 is estimated by the value of the denominator of Equation (45), when I0N ! 1, which is inﬁnite. To circumvent this problem, we follow the approach of Giorgilli and Skokos (1997), namely, we ﬁnd the maximum of (45) with respect to I0N , which corresponds to the least possible underestimate of the Nekhoroshev time. The maximum of (45) is at I0N ¼ ðN þ 1ÞIN =ðN À 1Þ, which yields the optimal Nekhoroshev time FORMAL INTEGRALS AND NEKHOROSHEV STABILITY tnk ¼ 2ð1 À AÞðNþ1Þ=2 ðNþ1Þ ðN þ 1ÞBqÃ jjUNþ1 jjðI0N ÞðNÀ1Þ=2 : 41 ð46Þ Solving Equation (46) with respect to I0N yields the level curve UN ðz; zÃÞ ¼ IN such that no initial condition in the interior of this curve may travel a distance larger than DIN ¼ I0N À IN within the time tnk .