Algebraic structure of the manifold of solutions of the $N$-body problem
Author:
Lawrence Goldman
Journal:
Proc. Amer. Math. Soc. 25 (1970), 417-422
MSC:
Primary 12.80; Secondary 70.00
DOI:
https://doi.org/10.1090/S0002-9939-1970-0260714-4
MathSciNet review:
0260714
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Abstract | References | Similar Articles | Additional Information
Abstract: The Theorem of Ritt on the decomposition of the perfect differential ideal generated by a single irreducible differential polynomial is, here, generalized to system of polynomials satisfying certain conditions. We use these results to prove that all solutions of the $N$-body problem, excepting the solutions for which one or more of the ${r_{ij}}$ (the distance between the masses ${M_i},\;{M_j}$) is zero, belong to one irreducible manifold.
- W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Book I: Algebraic preliminaries; Book II: Projective space; Reprint of the 1947 original. MR 1288305
- E. Leimanis, Some recent advances in the dynamics of rigid bodies and celestial mechanics. Dynamics and nonlinear mechanics, Surveys in Applied Mathematics, Vol. 2, John Wiley & Sons, Inc., New York Chapman & Hall, Ltd., London, 1958, pp. 1–108. MR 0096393
- Joseph Fels Ritt, Differential Algebra, American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, N. Y., 1950. MR 0035763
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Keywords:
Manifold of systems,
decomposition of radical ideals,
decomposition of perfect differential ideals,
Jacobian,
<IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$N$">-body problem,
three-body problem
Article copyright:
© Copyright 1970
American Mathematical Society