The existence of global solution of the n-body problem

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Based on McGehee's transformation f = I-1q; g = I 1 2p;, dt dt′ = I 3 2, I introduce the transformtion X = (gTM-1g) 1 2, h ≥ 0X = (gTM-1g) - 2Ih) 1 2, h≤0, u = IX-2; F = X2f; G = X-1g; dt′ dt = X-3. I prove that these variables may be continued to every point of the new time axis t for any initial value, and the whole axis corresponds to the "time interval of existence of the global solution". Also, F, G, u are O(e). I then obtain a region H on the complex plane τ, |Im(τ)| < A exp (B Re τ2), over which F and G are analytic. Here, A, B, C are constants related only to the masses and the initial value. Lastly, a conformal mapping is established which maps a subregion of H, H, onto the unit circle of the new complex variable, thus obtaining a global solution of the n-body problem. The convergence of my power series is admittedly unsatisfactory and so the present result is of limited value for practical calculation.

Original languageEnglish (US)
Pages (from-to)135-142
Number of pages8
JournalChinese Astronomy and Astrophysics
Issue number2
StatePublished - Jun 1986

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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