Abstract
This paper is a survey on the author’s work on the non-collision singularities in the Newtonian n-body problem. The non-collision singularity in the n-body system corresponds to the solution which blow up to infinity in finite time. The question whether there exists such solution was first raised by Painleve in the last century and since then, it has been open. Here, we show that such solutions do exist in a 5-body problem. The method we use is based on careful analysis of near collisions orbits and McGehee’s technique of blowing up collision singularities.
Partially supported by an NSF grant and Alfred Sloan Fellowship.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Devaney, Triple collision in the planar isosceles three-body problem, Inv. Math. 60 (1980), 249–267.
R. Devaney, Singularities in classical mechanical systems, Ergodic Theory and Dynamical Systems (1981), Birkhauser, Boston.
J. Mather and R. McGehee, Solutions of the collinear four body problem which become unbounded in finite time, Lecture Notes in Physics 38 (J. Moser, ed.) (1975), 573–597, Springer-Verlag, Berlin Heidelberg New York.
R. McGehee, Triple collision in the collinear three-body problem, Inventiones Math. 27 (1974), 191–227.
R. McGehee, Von Zeipe Vs theorem on singularities in celestial mechanics, Expo. Math. 4 (1986), 335–345.
R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SI AM J. Math. Analysis 15 (1984), 857–876.
P. Painlevé, “Leçons sur la théorie analytique des équations différentielles,” A. He- mann, Paris, 1897.
H. Pollard and D. Saari, Singularities of the n-body problem, I, Arch. Rational Mech. and Math. 30 (1968), 263–269.
H. Pollard and D. Saari, Singularities of the n-body problem, II, Inequalities-II (1970), 255–259, Academic Press.
D. Saari, Singularities and collisions of Newtonian gravitational systems, Archive for Rational Mechanics and Analysis 49, no. 4 (1973), 311–320.
D. Saari, Singularities of Newtonian gravitational systems, Proceedings of Symposium on Global Analysis, Dynamical Systems and Celestial Mechanics (1971), Brazil, August.
Z. Xia, The Existence of Non-collision Singularities In Newtonian System, Thesis (1988), Northwestern University.
H. von Zeipel, Sur les singularités du problème des n corps, Arkiv for Matematik, Astronomi och Fysik 4, 32 (1908), 1–4.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Xia, Z. (1991). The Non-collision Sigularities of the 5 body Problem. In: Ratiu, T. (eds) The Geometry of Hamiltonian Systems. Mathematical Sciences Research Institute Publications, vol 22. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9725-0_19
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9725-0_19
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9727-4
Online ISBN: 978-1-4613-9725-0
eBook Packages: Springer Book Archive