Permanent configurations in the problem of four bodies

MacMillan, William D.; Bartky, Walter

doi:10.1090/s0002-9947-1932-1501666-7

Cited by 78 publications

(37 citation statements)

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“…Proposition 2. Let A(s) and F (s) be a matrix and determinant of the form (9) with s ij arbitrary complex numbers. Suppose F (s) = 0 but at least one of the cofactors…”

Section: Geometry Of Configurationsmentioning

confidence: 99%

Generic Finiteness for Dziobek Configurations

Moeckel

2001

Trans. Amer. Math. Soc.

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Abstract. The goal of this paper is to show that for almost all choices of n masses, m i , there are only finitely many central configurations of the Newtonian n-body problem for which the bodies span a space of dimension n − 2 (such a central configuration is called a Dziobek configuration). The result applies in particular to two-dimensional configurations of four bodies and threedimensional configurations of five bodies.

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“…Proposition 2. Let A(s) and F (s) be a matrix and determinant of the form (9) with s ij arbitrary complex numbers. Suppose F (s) = 0 but at least one of the cofactors…”

Section: Geometry Of Configurationsmentioning

confidence: 99%

Generic Finiteness for Dziobek Configurations

Moeckel

2001

Trans. Amer. Math. Soc.

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“…For n ≥ 2 the equations of motion of the planar n-body problem ( [1], [2], [3], [5], [6], [7]) can be written in the formz…”

Section: Resultsmentioning

confidence: 99%

Periodic solutions for planar 2N-body problems

Zhang

Zhou

2002

Proc. Amer. Math. Soc.

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“…In [ 11], MacMillan and Bartky also show that for (cr, A) e 9?, A is uniquely determined by cr except in the case that cr represents the following configuration: three particles form an equilateral triangle with the fourth at the centre. So except in this case, the configuration determines the mass ratios uniquely.…”

Section: Is Bounded Onmentioning

confidence: 94%

“…To study this question it is desirable to transform (1.3) to make the mass m appear in the simplest possible way. To this end we define (following [11] Returning to the anti-symmetric matrix A we see that co is degenerate if and only if det (A) = 0. This leads one to suspect a connection between pf (to) and det (A).…”

Section: The Set Of All Relative Equilibriamentioning

confidence: 99%

“…Another approach to the subject is to ask which position vectors z are relative equilibria for some m, and then, given such a relative equilibrium, to find all of the corresponding mass vectors. Such a study was carried out by Dziobek [3] and by W. D. MacMillan and W. Bartky [11] in the case N = 4 and by W. L. Williams [28] in the case N = 5. Among the results are beautiful characterizations of the set of all non-collinear relative equilibria by explicit equations and uniqueness theorems for the corresponding mass vectors.…”

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confidence: 99%

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Relative equilibria of the four-body problem

Moeckel

1985

Ergod. Th. Dynam. Sys.

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Abstract. 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. Introduction; relative equilibria of the N-body problemThe Newtonian TV-body problem concerns the motion of TV point masses under the influence of their mutual gravitational attraction. We will consider the case when all the particles move in a fixed plane which we take to be the complex plane, C. The position vector zeC N is the vector (z,, ...,z n ) where z,-= x,-+ iy,-£ C is the position of the yth particle. The mass vector m e R n+ is (m l ,..., m N ) where m, is the mass of the jth particle. In complex notation, Newton's laws are:

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Permanent configurations in the problem of four bodies

Cited by 78 publications

References 0 publications

Generic Finiteness for Dziobek Configurations

Generic Finiteness for Dziobek Configurations

Periodic solutions for planar 2N-body problems

Relative equilibria of the four-body problem

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