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A minimizing property of Lagrangian solutions. (English) Zbl 0988.70007
This short paper shows that Lagrangian solutions of three-body problem minimize the action functional. The main result states that any minimal regular solution of the three-body problem is precisely the Lagrangian elliptic solution. The authors also give a brief review of Keplerian orbits and of the Gordon’s result.
MSC:
70F07Three-body problems
References:
[1]R. Abraham, J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings, 1978
[2]W. Gordon, A minimizing property of Keplerian orbits, Amer. J. of Math., 1977, 99:961–971 · Zbl 0378.58006 · doi:10.2307/2373993
[3]H. Pollard, Celestial Mechanics, The Mathematical Association of America, 1976
[4]J. Lagrange, Essai sur le probléme des trois crops, 1772, Ouvers, 1783, 3:229–331
[5]A. Chenciner, N. Desolneux, Minima de l’intégrale d’action etéquilibres relatifs de n corps, C. R. Acad. Sci. ParisI, serie, t.32, 1998, 1209–1212
[6]Y. Long, S. Q. Zhang, Geometric characterization for variational minimization solutions of the 3-body problem, Chinese Sci. Bull., 1999, 44:1653–1655 · Zbl 02065817 · doi:10.1007/BF03183482
[7]C. Siegel, J. Moser, Lectures on Celestial Mechanics, Berlin:Springer, 1971