Simple Choreographies of the Planar Newtonian N-Body Problem

Yu, Guowei

doi:10.1007/s00205-017-1116-1

Cited by 26 publications

(44 citation statements)

References 29 publications

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“…For one reason, Marchal's average method or the rotating circle property does not work, and in some cases it has been proven that an action minimizer does contain collision, see [19], [29] and [22]. Because of this, fewer results are available along this line, see [4], [5], [7], [18], [26], [30] and [33].…”

Section: Introductionmentioning

confidence: 99%

Spatial Double Choreographies of the Newtonian 2n-Body Problem

2018

Arch Rational Mech Anal

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Abstract. In this paper, for the spatial Newtonian 2n-body problem with equal masses, by proving the minimizers of the action functional under certain symmetric, topological and monotone constraints are collision-free, we found a family of spatial double choreographies, which have the common feature that half of the masses are circling around the z-axis clockwise along a spatial loop, while the motions of the other half masses are given by a rotation of the first half around the x-axis by π.Both loops are simple, without any self-intersection, and symmetric with respect to the xz-plane and yz-plane. The set of intersection points between the two loops is non-empty and contained in the xy-plane. The number of such double choreographies grows exponentially as n goes to infinity.

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Section: Introductionmentioning

confidence: 99%

Spatial Double Choreographies of the Newtonian 2n-Body Problem

2018

Arch Rational Mech Anal

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“…The next lemma follows from [1, Theorem 2.1]. In the following, for i = 1, 2, let η i (τ ), τ ∈ R, be two solutions of the linear system (28) given in (34) and (35), and V (τ ) = span{η 1 (τ ), η 2 (τ )} is defined in Definition 3.1. Since x(t) is non-homothetic, By Lemma 3.2, V (τ ) ∈ C 0 (R, Lag(R 4 )).…”

Section: Connect the Morse And Oscillation Indices By Maslov Indicesmentioning

confidence: 99%

“…We believe our result could be useful in deepening the variational study of the singular Lagrange systems including the classic N -body problem. In recent years, many new periodic and quasi-periodic solutions have been found as collision-free minimizers in the N -body problem under symmetric and/or topological constraints (see [14], [19], [13], [35]). However no result is available through minimax methods due to the problem of collision.…”

Section: Introductionmentioning

confidence: 99%

An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

2018

Commun. Math. Phys.

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In the variational study of singular Lagrange systems, the zero energy solutions play an important role. In this paper we find a simple way of computing the Morse indices of these solutions for the planar anisotropic Kepler problem. In particular an interesting connection between the Morse indices and the oscillating behaviors of these solutions discovered by the physicist M. Gutzwiller is established.

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“…One goal of this paper is to show the existence of prograde double-double orbits for an interval of θ. Instead of using local deformation argument [1,2,9,15], we introduce simple topological constraints to a two-point free boundary value problem and apply a level estimate argument to exclude possible collisions in the corresponding minimizers. Let the masses be m 1 = m 2 = m 3 = m 4 = 1.…”

Section: As Inmentioning

confidence: 99%

Existence of Prograde Double-Double Orbits in the Equal-Mass Four-Body Problem

Kuang

Yan

2018

Advanced Nonlinear Studies

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By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde doubledouble orbits for any rotation angle θ ∈ (0, π/7] in the equal-mass four-body problem.A new geometric argument is introduced to show that for any θ ∈ (0, π/2), the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

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Simple Choreographies of the Planar Newtonian N-Body Problem

Cited by 26 publications

References 29 publications

Spatial Double Choreographies of the Newtonian 2n-Body Problem

Spatial Double Choreographies of the Newtonian 2n-Body Problem

An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

Existence of Prograde Double-Double Orbits in the Equal-Mass Four-Body Problem

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