Hip-hop solutions of the 2N-body problem

Barrabés, Esther; Cors, Josep M.; Pinyol, Conxita; Soler, Jaume

doi:10.1007/s10569-006-9016-y

Cited by 20 publications

(34 citation statements)

References 9 publications

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“…Other symmetries are possible (see for instance [2]). We do not intend to explore, in the present paper, all possible symmetries but, indeed, to show that for high eccentricities hip-hop solutions can be shown to exist by means of topological arguments.…”

Section: Proposition 2 Let Q(t) = (R(t) D(t)ṙ(t)ḋ(t)) Be a Solutiomentioning

confidence: 99%

“…The orthogonal projection of both N -gons on the z = 0 plane will always be a regular rotating 2N -gon of variable size. A hip-hop solution is a periodic solution of this type, where periodic has the usual meaning of periodic in an ad-hoc rotating reference frame (see [2], [6]). …”

Section: Introductionmentioning

confidence: 99%

“…With these methods it is possible to find solutions that do not depend on a small parameter (see [3], [6] and the references therein for more details). In [2], the authors show that Poincare's argument of analytic continuation can be used to add vertical oscillations to the circular motion of 2N bodies of equal mass occupying the vertices of a regular 2N -gon, and prove the existence of families of hip-hop solutions with eccentricity close to zero. An infinite number of these orbits are 3D choreographies, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…The study of hip-hop solutions is reduced to the study of the three-degree of freedom system given in Proposition 1 (see [2] for details). r), where R is the matrix given in (2), is a solution of the 2N -body problem (1) if and only if r(t) satisfies the equation…”

mentioning

confidence: 99%

“…r), where R is the matrix given in (2), is a solution of the 2N -body problem (1) if and only if r(t) satisfies the equation…”

mentioning

confidence: 99%

See 4 more Smart Citations

Highly eccentric hip–hop solutions of the 2–body problem

Barrabés

Cors

Pinyol

et al. 2010

Physica D: Nonlinear Phenomena

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We show the existence of families of hip-hop solutions in the equal-mass 2N -body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non-harmonic oscillations. By introducing a parameter , the homographic motion and the small amplitude oscillations can be uncoupled into a purely Keplerian homographic motion of fixed period and a vertical oscillation described by a Hill type equation. Small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise, via a Bolzano argument, to resonant periodic solutions of the uncoupled system in a rotating frame. For small = 0, the topological transversality persists and Brouwer's fixed point theorem shows the existence of this kind of solutions in the full system.

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Section: Proposition 2 Let Q(t) = (R(t) D(t)ṙ(t)ḋ(t)) Be a Solutiomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…r), where R is the matrix given in (2), is a solution of the 2N -body problem (1) if and only if r(t) satisfies the equation…”

mentioning

confidence: 99%

See 3 more Smart Citations

Highly eccentric hip–hop solutions of the 2–body problem

Barrabés

Cors

Pinyol

et al. 2010

Physica D: Nonlinear Phenomena

Self Cite

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Four lectures on the N-body problem

Chenciner¹

Hamiltonian Dynamical Systems and Applications

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Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

Calleja

Doedel

García-Azpeitia

2018

Celest Mech Dyn Astr

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We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases n = 4, 6, 7, 8, and 9. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link below 1 .

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Hip-hop solutions of the 2N-body problem

Cited by 20 publications

References 9 publications

Highly eccentric hip–hop solutions of the 2–body problem

Highly eccentric hip–hop solutions of the 2–body problem

Four lectures on the N-body problem

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

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