Perturbing the Lagrange Solution to the General Three Body Problem

Piña, E.; Jiménez-Lara, Lidia

doi:10.1007/978-1-4419-9058-7_8

Cited by 4 publications

(4 citation statements)

References 3 publications

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“…The observation that the second equation in (2.6) implies to choose the origin of σ at the constant value demanded by the Lagrange solution was discovered in reference [13]. The proof is there.…”

Section: Comments and Proofsmentioning

confidence: 87%

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Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

Piña

Bengochea

2009

Qual. Theory Dyn. Syst.

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When the gravitational three-body problem in the plane is studied in the Piña and Jiménez system of coordinates, the shape sphere is simply related to the coordinates. This shape sphere is considered as a four 2-sphere, where contrary to other authors, the poles of the sphere occur for two equal inertia moments, not for the Lagrange points. In this sphere the central configurations correspond to fixed points of the sphere which are functions of the three masses. The binary collisions are also represented as points in this sphere. The hyperbolic geometry of the collision angles is the link between the instantaneous moments of inertia and the distances between particles.

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Section: Comments and Proofsmentioning

confidence: 87%

“…[13]. In this reference we write only a different factor 2 between a component of the third column of matrix C and the corresponding component of the third row of matrix C −1 .…”

Section: Comments and Proofsmentioning

confidence: 98%

Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

Piña

Bengochea

2009

Qual. Theory Dyn. Syst.

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“…In [4] and [5], the condition is bM 2 a T = 0, and in [1] and [2] the extra condition is a · b = ab T = 0, where the dot is the usual Euclidean product. Once a basis is chosen, the collision angles are fixed.…”

Section: The Planar Three-body Problemmentioning

confidence: 99%

The Planar Three-Body Problem, Symmetries and Periodic Orbits

Prada

Jiménez-Lara

2009

Qual. Theory Dyn. Syst.

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We study the planar three-body problem using the principal axes of inertia frame, determined by one Euler angle in the planar case. Three variables are needed in this frame: two distances R 1 and R 2 , related with the principal moments of inertia on the plane of motion of the particles, and an auxiliary angle σ . We also connect these coordinates with the shape sphere of the similarity class of triangles and we give a geometric interpretation of the angle σ . We then write the Hamiltonian and equations of motion in these coordinates to find the symmetries of involution and symmetric periodic orbits. To exemplify this method, we calculate periodic orbits of a three-body problem with two or three equal masses.

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“…al. en un sistema de coordenadas apropiado [31,51,53,54,55]. Estos consisten en unángulo de Euler ψ que rota del sistema inercial al sistema de ejes principales de las partículas, siendo el tercer eje el que es perpendicular al plano, dos distancias R 1 y R 2 relacionadas a los momentos de inercia principales mediante I j = µR 2 j , j = 1, 2, con µ una masa constante, así como unángulo σ, que lleva del sistema de ejes principales a uno fijo apropiado.…”

Section: Introductionunclassified

Órbitas periódicas en el problema de tres cuerpos

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Perturbing the Lagrange Solution to the General Three Body Problem

Cited by 4 publications

References 3 publications

Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

The Planar Three-Body Problem, Symmetries and Periodic Orbits

Órbitas periódicas en el problema de tres cuerpos

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