The global solution of the n-body problem. (English) Zbl 0726.70006

Summary: The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of \(n=3\) with nonzero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of \(n>3\) or to \(n=3\) with zero-angular momentum.
A new ‘blowing up’ transformation, which is a modification of McGehee’s transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of \(n>3\) and \(n=3\) with zero angular momentum.


70F10 \(n\)-body problems
70F15 Celestial mechanics
Full Text: DOI


[1] Siegel, C.L. and Moser, J.K.: 1971, Lecture on Celestial Mechanics, Springer-Verlag. · Zbl 0312.70017
[2] Sundman, K.F.: 1913, ?Mémoire sur le problème des trois corps?, Acta Math. 36, 105-179. · JFM 43.0826.01
[3] McGehee, R.: 1974, ?Triple collision in the collinear three-body problem?, Invent. Math. 27, 191-277. · Zbl 0297.70011
[4] Mather, J.N. and McGehee, R.: 1974, ?Solution for collinear four-body problem which becomes unbounded in finite time?, Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst. Seattle, Wash.) pp. 573-597.
[5] Pollard, H.: 1967, ?Gravitational system?, J. Math. 17, 601-612. · Zbl 0159.26102
[6] Saari, D.G.: 1971, ?Expanding gravitational system?, Trans. Amer. Math. Soc. 156, 219-240. · Zbl 0215.57001
[7] Marchel, C. and Saari D.G.: 1976, ?On the final evolution of n-body problem?, J. Dif. Eq. 20.
[8] Saari, D.G.: 1971, 1973, ?Improbability of collision in Newtonian gravitational system?, (I) Trans. Amer. Math. Soc. 162, 267-271. (II) ibid. 181, 361. · Zbl 0231.70007
[9] Saari, D.G.: 1971, ?Singularities and collision in Newtonian graviational system?, Arch. Rat. Mech. and Anal. 49, 311-320.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.