PERIODIC OSCILLATIONS IN THE RESTRICTED HIP-HOP 2N1-BODY PROBLEM ANDR ES RIVERA1 OSCAR PERDOMO2 NELSON CASTA NEDA2.

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PERIODIC OSCILLATIONS IN THE RESTRICTED HIP-HOP 2N+1-BODY

PROBLEM

ANDR´

ES RIVERA 1, OSCAR PERDOMO 2, NELSON CASTA ˜

NEDA 2.

Abstract. We prove the existence of periodic solutions of the restricted (2N+1)-body

problem when the 2N-primaries move on a periodic Hip-Hop solution and the massless body

moves on the line that contains the center of mass and is perpendicular to the base of the

antiprism formed by the 2N-primaries.

1. Introduction.

In a classical restricted (n+1)-body problem the motion of one body (that we call the plus

one body) with mass m0≈0 is aﬀected only by the gravitational force of the other n-bodies

and does not perturb the motion of the n-bodies, which interact only with gravitational

forces. When the n-bodies have the same mass m>0 they are usually called primaries. This

mathematical model can be used to describe motions of comets, spacecraft and asteroids (see

[17] and the references therein). In the literature, there is a remarkable example of a restricted

3-body problem called the Sitnikov problem (proposed in 1960 by K.A. Sitnikov [18]). Here

the two primaries move in elliptic orbits lying in the xy-plane around the center of mass

(barycenter) as solutions of the 2-body problem. Finally, the plus one body moves along the

z-axis passing through the barycenter of the primaries. The Sitnikov problem deals with the

orbits of the plus one body. Since its formulation, this particular 3-body problem has been

treated in several papers [1,4,6,7,9,10,12,16,18] from both, numerical and analytical point

of view. Concerning the existence of periodic motions for the plus one body in the Sitnikov

problem, we refer to [10,12] where the authors prove the existence of families of symmetric

periodic orbits which depend continuously on the eccentricity of the primaries, by means of

the Leray-Schauder continuation method. Using the same technique, in [15] the author proves

that these families also exist in a particular generalization of the Sitnikov problem where the

number of primaries is n≥2. More precisely, in [15] the authors ﬁnd the existence of periodic

motions in a restricted (n+1)-body problem where each primary body is at the vertex of a

regular n-gon and moves on elliptic orbits lying in the xy-plane around their barycenter and

the plus one body moves on the z-axis. The link https://youtu.be/RjlZpqDFsDM leads to

a video showing some of these periodic motions when the plus one body remains still at the

2010 Mathematics Subject Classiﬁcation. 70F10, 37C27, 34A12.

Key words and phrases. N-body problem, periodic orbits, hip-hop solutions, implicit function theorem,

bifurcations.

arXiv:2210.01740v1 [math.DS] 4 Oct 2022

2 ANDR´

ES RIVERA, OSCAR PERDOMO, NELSON CASTA ˜

NEDA

origin (the center of mass). Moreover, when the eccentricity of orbits is zero, all the primaries

have the same circular orbit, i.e., the primaries perform a choreography.

In this paper we are considering also n-primaries but this time they do not move on a plane,

they are periodic Hip-hop solutions of the n-body problem. The existence of these solutions

were studied in [2,3] and more recently, in [13] we found the existence of families of periodic

Hip-hop solutions for a 2N-body problem. A hip-hop solution satisﬁes i)All the bodies have

the same mass. ii)at every instante of time t,Nof the bodies are at the vertices of a regular

N-gon contained in a plane Π1(t)and the other N-bodies are at the vertices of a second regular

N-gon contained in a plane Π2(t)which is obtained from the ﬁrst N-gon by the reﬂection in

a ﬁxed plane Π0followed by a rotation of πNradians around a ﬁxed-line l0perpendicular

to the three planes and passing through the center of the two N-gons. When Π1(t)≠Π2(t),

the 2N-bodies are at the vertices of an antiprism that degenerates to a regular 2N-gon when

Π1(t)=Π2(t). At any time t, the oriented distance from the plane Π0to Π1(t)is given by a

function d(t)and the distance of any primary body to the line l0is given by a function r(t).

Without loss of generality, it is possible to assume that the line l0is the z-axis and the plane

Π0is the xy-plane. Let us explain in more detail the new type of restricted (2N+1)-body

problem which can be thought of as a sort of Sitnikov problem.

The restricted hip-hop body problem. Consider 2N-bodies with equal mass (called

primaries) located at any time at the vertices of a regular antiprism, moving as a periodic

Hip-hop solution of a 2N-body problem and a massless body moving on the straight line

orthogonal to the two regular N-gons that form the antiprism and passes through their center

of mass. The restricted hip-hop body problem will consist in describing the motion of the

massless body.

Figure 1. Restricted hip-hop body problem with four primaries. At any time,

the primaries are placed at the vertices of a regular 4-gonal antiprism and the

plus one body moves on the line orthogonal to the two 2-gons that are the base

face of the antiprism.

HIP-HOP SOLUTIONS 3

The main objective of this paper is to analytically prove the existence of symmetric periodic

orbits for the restricted antiprism body problem. To this end, the paper is divided as follows.

In Section 2we deduce the equation of motion for the 2N-primaries and the plus one body. In

Section 3, we apply the same techniques found in [13] by reducing the problem of the existence

of periodic solutions to the problem of solving a system of three equations in four variables.

The main results of the paper are contained in Section 4and some numerical periodic solutions

of the restricted hip-hop problem for 6 primaries are given in Section 5.

2. The differential equations

We will follow the same set up as in the references [13] and [2]. To describe the motion of the

primaries we consider Q1, Q2,...,Q2Nbodies with equal mass m>0, located on the vertices

of a regular anti-prism. If rj(t)is the position of the body Qj, j =1,...,2Nat each instant t

then

rj(t)=Rj−1r1(t), j =1,...,2N,

where r1(t)=(r(t)cos(θ(t)), r(t)sin(θ(t)), d(t))and Ris a rotation/reﬂection matrix given

R=





cos(π

N)−sin(π

N)0

sin(π

N)cos(π

N)0

0 0 −1



.

We have that the functions r(t),θ(t)and d(t)that deﬁne r1(t)satisfy the equations











¨r=f(r, d),

d=g(r, d),

θ=ar2,

(1)

where ais the angular momentum of the system and

f(r, d)=a2

r3−2rm

2N−1

∑

k=1

sin2(kπ2N)

4r2sin2(kπ2N)+((−1)k−1)2d23/2,

g(r, d)=−m d

2N−1

∑

k=1((−1)k−1)2

4r2sin2(kπ2N)+((−1)k−1)2d23/2.

Now that we have a description of the 2Nprimaries, we describe the motion of the plus

one body. Since this body moves on the z-axis, then its motion is describedby the map

r2N+1(t)=(0,0, z(t)). Notice that the equation of the primaries do not change because of our

assumption, also we notice in this model the mass of the plus one body is irrelevant because

it cancels out from the equations.

A direct computation shows that z(t)satisﬁes the following diﬀerential equation.

¨z=h(r, d, z)=−mN z−d

((z−d)2+r2)3/2+z+d

((z+d)2+r2)3/2(2)

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摘要：

PERIODICOSCILLATIONSINTHERESTRICTEDHIP-HOP2N1-BODYPROBLEMANDRESRIVERA1,OSCARPERDOMO2,NELSONCASTA~NEDA2.Abstract.Weprovetheexistenceofperiodicsolutionsoftherestricted2N1-bodyproblemwhenthe2N-primariesmoveonaperiodicHip-Hopsolutionandthemasslessbodymovesonthelinethatcontainsthecenterofmassandispe...

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PERIODIC OSCILLATIONS IN THE RESTRICTED HIP-HOP 2N1-BODY PROBLEM ANDR ES RIVERA1 OSCAR PERDOMO2 NELSON CASTA NEDA2.

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