Finiteness in the Planar Restricted Four-Body Problem

Kulevich, Julianne L.; Roberts, Gareth E.

doi:10.1007/s12346-010-0006-9

Cited by 17 publications

(13 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The boundaries of this surface correspond to three important symmetric cases: a kite, an isosceles trapezoid, and a degenerate case where three bodies lie at the vertices of an equilateral triangle and the fourth body of the quadrilateral has zero mass. This last case corresponds to central configurations of the planar, circular, restricted four-body problem [13]. We also prove that for any c.c.c., the masses must be ordered in a precise fashion, with the largest body lying on the vertex between the two longest exterior sides, the smallest body opposite the largest, and the two largest (smallest, respectively) bodies lying on the longest (smallest, resp.)…”

Section: Introductionmentioning

confidence: 77%

Four-body co-circular central configurations

Cors

Roberts

2012

Nonlinearity

View full text Add to dashboard Cite

We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.MSC Classifications: 70F10, 70F15, 37N05

show abstract

Section: Introductionmentioning

confidence: 77%

Four-body co-circular central configurations

Cors

Roberts

2012

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…In the former case, θ = π/2 and then r 12 = r 23 and r 14 = r 34 follows quickly. The latter case is impossible, since c = 0 and b = 1 contradicts inequality (20). This proves (34).…”

Section: Configurations On the Boundary Of Dmentioning

confidence: 61%

“…The combined inequalities between the radial variables a, b, and c given in (15) through (20), along with a > 0, b > 0, and c > 0, define a bounded set D ⊂ R + 3 . We will show that this set is the domain of the function θ = f (a, b, c) and the projection of E into abc-space.…”

Section: Defining the Domain Dmentioning

confidence: 99%

Classifying Four-Body Convex Central Configurations

Corbera,

Cors,

Roberts

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain D that is the union of elementary regions in R + 3 .

show abstract

“…For instance, in [35, p. 168] we find that for some choices of the masses the number of libration points can be just 8, while in [2, p. 14] it is announced that the number of libration points can be 8, 9 or 10 depending on the masses. Some geometrical insight in the nineties [24, §III] has been followed by a number of computer-assisted proofs in the new millennium [3,17,18], which have given a renewed interest to the problem.…”

Section: The Restricted Triangular 4-body Problemmentioning

confidence: 99%

Bifurcation of closed orbits from equilibria of Newtonian systems with Coriolis forces

Gołȩbiewska,

Pérez-Chavela,

Rybicki

et al. 2021

Preprint

View full text Add to dashboard Cite

We consider autonomous Newtonian systems with Coriolis forces in two and three dimensions and study the existence of branches of periodic orbits emanating from equilibria. We investigate both degenerate and nondegenerate situations. While Lyapunov's center theorem applies locally in the nondegenerate, nonresonant context, equivariant degree theory provides a global answer which is significant also in some degenerate cases.We apply our abstract results to a problem from Celestial Mechanics. More precisely, in the three-dimensional version of the Restricted Triangular Four-Body Problem with possibly different primaries our results show the existence of at least seven branches of periodic orbits emanating from the stationary points.

show abstract

Finiteness in the Planar Restricted Four-Body Problem

Abstract: Using BKK theory, we show that the number of equilibria (central configurations) in the planar, circular, restricted four-body problem is finite for any choice of masses. Moreover, the number of such points is bounded above by 196.

Cited by 17 publications

References 12 publications

Four-body co-circular central configurations

Four-body co-circular central configurations

Classifying Four-Body Convex Central Configurations

Bifurcation of closed orbits from equilibria of Newtonian systems with Coriolis forces

Contact Info

Product

Resources

About