Relative equilibria of the four-body problem. (English) Zbl 0554.14004
By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.
MSC:
| 14E05 | Rational and birational maps |
| 14Pxx | Real algebraic and real-analytic geometry |
| 70F10 | \(n\)-body problems |
Keywords:
bifurcation of relative equilibria; Newtonian 4-body problem; algebraic correspondence between real algebraic varietiesReferences:
| [1] | DOI: 10.1007/BF01230241 · Zbl 0422.70014 · doi:10.1007/BF01230241 |
| [2] | Siegel, Lectures on Celestial Mechanics (1971) · doi:10.1007/978-3-642-87284-6 |
| [3] | DOI: 10.1007/BF02404404 · JFM 47.0837.01 · doi:10.1007/BF02404404 |
| [4] | Lehmann-Filhes, Astron. Nach. 127 pp none– (1891) |
| [5] | Lagrange, Ouvres 6 pp 272– (1873) |
| [6] | Hoppe, Archiv der Math, und Phys. 64 pp 218– (none) |
| [7] | Greub, Multilinear Algebra pp 200– (1978) · doi:10.1007/978-1-4613-9425-9 |
| [8] | Euler, Novi Comm. Acad. Sci. Imp. Petrop. 11 pp 144– (1767) |
| [9] | DOI: 10.1002/asna.19001520302 · doi:10.1002/asna.19001520302 |
| [10] | Wintner, The Analytical Foundations of Celestial Mechanics 5 (1941) · JFM 67.0785.01 |
| [11] | Andoyer, Bull. Astron. 25 pp 50– (1906) |
| [12] | DOI: 10.2307/1989896 · Zbl 0020.31901 · doi:10.2307/1989896 |
| [13] | DOI: 10.2307/1969908 · Zbl 0078.13403 · doi:10.2307/1969908 |
| [14] | DOI: 10.2307/1970964 · Zbl 0321.58014 · doi:10.2307/1970964 |
| [15] | DOI: 10.1007/BF00405589 · Zbl 0328.57013 · doi:10.1007/BF00405589 |
| [16] | DOI: 10.1090/S0002-9904-1975-13794-3 · Zbl 0328.57012 · doi:10.1090/S0002-9904-1975-13794-3 |
| [17] | DOI: 10.1090/S0002-9904-1973-13254-9 · Zbl 0273.57016 · doi:10.1090/S0002-9904-1973-13254-9 |
| [18] | Mumford, Complex Projective Varieties pp 42– (1976) |
| [19] | DOI: 10.2307/2007159 · JFM 41.0794.02 · doi:10.2307/2007159 |
| [20] | DOI: 10.2307/2034050 · Zbl 0123.38302 · doi:10.2307/2034050 |
| [21] | DOI: 10.2307/1989432 · Zbl 0005.41802 · doi:10.2307/1989432 |
| [22] | DOI: 10.1090/S0002-9904-1907-01475-1 · JFM 38.0726.01 · doi:10.1090/S0002-9904-1907-01475-1 |
| [23] | Thorn, Differential and Combinational Topology pp 255– (1965) |
| [24] | DOI: 10.1007/BF01389805 · Zbl 0203.26102 · doi:10.1007/BF01389805 |
| [25] | DOI: 10.1007/BF01418778 · Zbl 0202.23201 · doi:10.1007/BF01418778 |
| [26] | DOI: 10.1007/BFb0068618 · doi:10.1007/BFb0068618 |
| [27] | DOI: 10.1007/BFb0068619 · doi:10.1007/BFb0068619 |
| [28] | DOI: 10.1007/BF01228714 · Zbl 0394.70009 · doi:10.1007/BF01228714 |
| [29] | Robbin, In Dynamical Systems (1973) · Zbl 0273.58005 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
