Rose solutions with three petals for planar 4-body problems

For some planar Newtonian N +3-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that N bodies chase each other on a curve, and the other 3 bodies chase each other on another curve. From the definition of the group action in equations (3.1) − (3.3), we can find that they are new solutions which are also different from all the examples of Ferrario and Terracini (2004) [22].

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“…Then by Lemma 2.2, f (q) attains inf {f (q)|q ∈ Λ2 }. Similar to Lemma 3.2 in [21], we can obtain the following lemma.…”

Section: Some Lemmasmentioning

confidence: 74%

New periodic solutions for some planar N+3-body problems with Newtonian potentials

Yuan

Zhang

2018

Journal of Geometry and Physics

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“…After the discovery of the first remarkable non-trivial choreographic solution -the figure eight of the three body problem by Moore (1993 [24]) and Chenciner and Montgomery (2000, [8]), many expertise attempt to study choreographic solutions and a large number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs. More results can be found in [2,5,6,9,10,12,14,15] and the reference therein.…”

mentioning

confidence: 89%

“…In the past decade, the existence of many new interesting periodic orbits are proved by using variational method for the n-body problem. Most of them are found by minimizing the Lagrangian action on a symmetric loop space with some topological constraints (for example, see [3,4,11,15,16,17,33,34]).…”

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confidence: 99%

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

Ouyang

Xie

2018

Journal of Differential Equations

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In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with m 1 = m 3 and m 2 = m 4 . A path q(t) in [0, T ] satisfies SPBC if the boundaries q(0) ∈ A and q(T ) ∈ B, where A and B are two structural configuration spaces in (R 2 ) 4 and they depend on a rotation angle θ ∈ (0, 2π) and the mass ratio µ = m2 m1 ∈ R + . We show that there is a region Ω ⊆ (0, 2π) × R + such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying SPBCThe corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with SPBC, we only impose constraints on boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions extended from the initial minimizing pathes have the symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when (θ, µ) = ( 4π 5 , 1). Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the basic equal mass choreography (See figure 1).

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“…Many expertises attempt to study choreographic solutions and a large number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs. More results can be found in [14][15][16]10,[17][18][19][20] and the reference therein.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem

Ouyang

Xie

2015

Physica D: Nonlinear Phenomena

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h i g h l i g h t s • Develop variational method with structural prescribed boundary conditions (SPBC). • This method numerically finds solutions and periodic solutions for N-body problem. • This method analytically proves the existence of such solutions numerically found. • Stable star pentagon choreography are analytically proved and numerically found. • Many stable choreographic and other periodic solutions are discovered and proved. Communicated by B. Sandstede Keywords: Variational method Choreographic periodic solutions Variational method with structural prescribed boundary conditions (SPBC) Stability Central configurations n-body problem a b s t r a c tIn this paper, we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical Newtonian 4-body problem. These orbits are discovered by a variational method with structural prescribed boundary conditions (SPBC). This method provides an initial path that is obtained by minimizing the Lagrangian action functional over the SPBC. We prove that the initial path can be extended to a periodic or quasi-periodic solution. With computer-assistance, a family of choreographic orbits of this type is shown to be linearly stable. Among the many linearly stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution. We also prove the existence of infinitely many double choreographic periodic solutions, infinitely many non-choreographic periodic solutions and uncountably many quasi-periodic solutions. Each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions. Our results with SPBC largely complement the current results by minimizing the action on a loop space.

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Rose solutions with three petals for planar 4-body problems

Cited by 5 publications

References 20 publications

New periodic solutions for some planar N+3-body problems with Newtonian potentials

New periodic solutions for some planar N+3-body problems with Newtonian potentials

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem

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