# Periodic Solutions for Circular RestrictedN+1-Body Problems

Authors: Xiaoxiao Zhao; Wei Li; Peng Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

For circular restrictedN+1-body problems, we study the motion of a sufficiently small mass point (called the zero mass point) in the plane of N equal masses located at the vertices of a regular polygon. By using variational minimizing methods, for some N, we prove the existence of the noncollision periodic solution for the zero mass point with some fixed wingding number.


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## Body

## 1. Introduction and Main Results

In this paper, we study the planar circular restrictedN+1-body problems. Suppose that points of positive masses m1=⋯=mN=1 move on a circular orbit around the center of masses, and that the sufficiently small mass point, called the zero mass point, moves on the moving plane of the given N equal masses and does not influence the motion of m1,…,mN, but the motion of the zero mass point is affected by the given N equal mass points. Let ρi=e-1(2πi/N) and we denote the position vectors of the given N bodies by qi(t), i=1,…,N, then
(1)q1(t)=re-12πtρ1,…,qi(t)=re-12πtρi,…,qN(t)=re-12πt,
where the radius r>0. It is known that q1(t),…,qN(t) satisfy the following Newtonian equations:
(2)miq¨i=∂U∂qi,i=1,…,N,
where
(3)U=∑1≤i<j≤N‍mimj|qi-qj|.
We also assume that
(4)∑i=1N‍miqi=∑i=1N‍qi=0.
The orbit q(t)∈R2 for the zero mass point is governed by the gravitational forces of m1,…,mN and therefore it satisfies the following equation:
(5)q¨=∑i=1N‍mi(qi-q)|qi-q|3.ForN-body problems, there are many papers concerned with the periodic solutions by using variational methods; see [1–15] and the references therein. In [1], Chenciner and Montgomery proved the existence of the remarkable figure-“8” type periodic solution for planar Newtonian 3-body problems with equal masses. Marchal [4] studied the fixed end problem for Newtonian n-body problems and proved that the minimizer for the Lagrangian action has no interior collision. In [6], Simó used computer to discover many new periodic solutions for Newtonian n-body problems. Zhang and Zhou [10–12] decomposed the Lagrangian action for n-body problems into some sum for two body problems and [11, 12] avoided collisions by comparing the lower bound for the Lagrangian action on the symmetry collision orbits and the upper bound for the Lagrangian action on test orbits in some cases.Motivated by the above works, we use variational methods to study the circular restrictedN+1-body problem with some fixed wingding numbers and N equal masses.For the readers’ conveniences, we recall the definition of the winding number, which can be found in many books on the classical differential geometry.Definition 1.
LetΓ:x(t), t∈[a,b] be a given oriented continuous closed curve, and p be a point on the plane not on the curve. Then, the mapping φ:Γ→S1 given by
(6)φ(x(t))=x(t)-p|x(t)-p|,t∈[a,b]
is defined to be the position mapping of the curve Γ relative to p, and when the point on Γ goes around the curve once, its image point φ(x(t)) will go around S1 a number of times; this number is called the winding number of the curve Γ relative to p, and we denote it by deg(Γ,p). If p is the origin, we write degΓ.Let(7)R(n)=(cos2πn-sin2πnsin2πncos2πn)∈SO(2)
be a counter-clockwise rotation of angle 2π/n in R2.Define(8)W1,2(RZ,R2)={x(t)∣x(t),x˙(t)∈L2(R,R2),x(t+1)=x(t)}.
The norm of W1,2(R/Z,R2) is
(9)∥x∥=[∫01‍|x|2dt]1/2+[∫01‍|x˙|2dt]1/2.
We consider the Lagrangian functional of (5) as
(10)f(q)=∫01‍[12|q˙|2+∑i=1N‍1|q-qi|]dt
on Λ±, where (11)Λ-={q∈W1,2(RZ,R2)|q(t+12)=R(2)q(t),q(t)≠qi(t),∀t∈[0,1],i=1,…,N,deg(q-qN)=-1,deg(qj-qN)=1,j=1,…,N-1},Λ+={q∈W1,2(RZ,R2)|q(t+1N)=R(N)q(t),q(t)≠qi(t),∀t∈[0,1],i=1,…,N,deg(q-qN)=1,deg(qj-qN)=1,j=1,…,N-1}.Our main results are the following.Theorem 2.
ForN=2,3, the minimizer of f(q) on the closure Λ¯- of Λ- is a noncollision 1-periodic solution of (5).Remark 3.
In proving Theorem2, we need to use test functions. We find that when N=3, if the test functions are circular orbits, we cannot get the desired results on Λ¯-. Therefore, we select elliptic orbits as test functions.Theorem 4.
For2≤N≤32402, the minimizer of f(q) on the closure Λ¯+ of Λ+ is a noncollision 1-periodic solution of (5).


## 2. Preliminaries

In this section, we will list some basic lemmas and inequality to prove our theorems.Lemma 5 (see [8]).
The radiusr for the moving orbit of N equal mass points is
(12)r=(14π)2/3[∑1≤j≤N-1csc(πNj)]1/3.Lemma 6 (Zhang and Zhou [9]).
Suppose thatg∈SO(k) is an element of finite order s and it has no fixed point other than the origin (i.e., 1 is not an eigenvalue of g). Then ∫0T‍X(t)dt=0 for all X(t)=(x1(t),…,xN(t))∈EgN, where EgN={(x1,…,xN)∣xi∈H1(R/TZ,RK),xi(t+T/s)=g(xi(t)),∀t∈[0,T],i=1,…,N}.Lemma 7 (Poincare-Wirtinger inequality [16]).
Letq∈W1,2(R/TZ,RK) and ∫0T‍q(t)dt=0, and then
(13)∫0T‍|q(t)|2dt≤T24π2∫0T‍|q˙(t)|2dt.Lemma 8 (see [17]).
LetX be a reflexive Banach space, S be a weakly closed subset of X, and f:S→R∪{+∞}. If f≢+∞ is weakly lower semicontinuous and coercive (f(x)→+∞ as ∥x∥→+∞), then f attains its infimum on S.Lemma 9 (Palais’s symmetry principle [18]).
Letσ be an orthogonal representation of a finite or compact group G, let H be a real Hilbert space, and let f:H→R satisfy f(σ·x)=f(x),∀σ∈G,∀x∈H.
SetF={x∈H∣σ·x=x,∀σ∈G}. Then the critical point of f in F is also a critical point of f in H.Remark 10.
By Palais’s symmetry principle and the perturbation invariance for wingding numbers, we know that the critical point off(q) in Λ± is a periodic solution of Newtonian equation (5).Lemma 11.
(1) (Gordon’s theorem [19]) Let x∈W1,2([t1,t2],RK) and x(t1)=x(t2)=0. Then for any a>0, one has
(14)∫t1t2‍(12|x˙|2+a|x|)dt≥32(2π)2/3a2/3(t2-t1)1/3.
(2) (Long and Zhang [20]) Let x∈W1,2(R/TZ,RK), and ∫0T‍xdt=0, then for any a>0, we have
(15)∫0T‍(12|x˙|2+a|x|)dt≥32(2π)2/3a2/3T1/3.


## 3. Proof of Theorems

In order to get our theorems, we need two steps to complete the proof.Step 1.
We will establish the existence of variational minimizers off(q) in (10) on Λ¯±.Lemma 12.
f
(
q
) in (10) attains its infimum on Λ¯±.Proof.
It is easy to check that the eigenvalueλ of R(n) satisfies
(16){λ=cos2πn,sin2πn=0,
which implies λ≠1 for n≥2. Then by Lemma 6, we have
(17)∫01‍q(t)=0∀q∈Λ¯±.
So by using Lemma 7, for all q∈Λ¯±, we see that an equivalent norm of (9) on Λ¯± is
(18)∥q∥≅[∫0T‍|q˙|2dt]1/2.
Hence, by the definition of f(q), f is coercive on Λ¯±.
Similar to the proof of Lemma 2.4 in [13], we can get the conclusion that f is weakly lower semi-continuous on Λ¯±.
Therefore, by using Lemma8, it can be concluded that f(q) in (10) attains its infimum on Λ¯±.Step 2.
We will prove that the variational minimizer in Lemma12 is the noncollision 1-period solution of (5).
For any collision generalized solutionq, we can estimate the lower bound for the value of Lagrangian action functional.Lemma 13.
For∂Λ-={q∈W1,2(R/Z,R2)∣q(t+1/2)=R(2)q(t), ∃1≤i0≤N, t0∈[0,1]s.t.qi0(t0)=q(t0)}, we have
(19)infq∈∂Λ-f(q)≥32(4π)2/3N-1/3+(N-1N)32(2π)2/3N2/3-12(2π)2r2≜d0.Proof.
It follows from (4) that
(20)∑i=1N‍q˙i=0,
which implies
(21)∑i=1N‍|q˙-q˙i|2=∑i=1N‍(|q˙|2+|q˙i|2-2〈q˙,q˙i〉)=N|q˙|2+∑i=1N‍|q˙i|2-2〈q˙,∑i=1N‍q˙i〉=N|q˙|2+∑i=1N‍|q˙i|2.
Therefore,
(22)|q˙|2=1N∑i=1N‍(|q˙-q˙i|2-|q˙i|2).
Hence,
(23)f(q)=∫01‍[12|q˙|2+∑i=1N‍1|q-qi|]dt=1N∫01‍∑i=1N‍[12|q˙-q˙i|2+N|q-qi|]dt-12N∫01‍∑i=1N‍|q˙i|2dt.
If q∈Λ¯- is a collision solution, then there exists t0∈[0,1] and 1≤i0≤N s.t q(t0)=qi0(t0). Since qi(t+1/2)=R(2)qi(t), one has q(t0+k/2)=qi0(t0+k/2),∀0≤k≤2. So, by (1) of Lemma 11, we get
(24)1N∫01‍[12|q˙-q˙i0|2+N|q-qi0|]dt=2N∫01/2‍[12|q˙-q˙i0|2+N|q-qi0|]dt≥32(4π)2/3N-1/3.
For noncollision pair q,qi(i≠i0), we claim ∫01‍(q(t)-qi(t))dt=0. In fact, it follows from (1) that qi+1(t)=qi(t+1/N), which implies
(25)qi(t)=q1(t+i-1N).
Therefore, by (4), we have
(26)∫01‍q1(t)dt=∫01/N‍q1(t)dt+⋯+∫(N-1)/N1‍q1(t)dt=∫01/N‍q1(t)dt+⋯+∫01/N‍q1(t+N-1N)dt=∫01/N‍∑i=1N‍qi(t)dt=0.
Since q1(t+1)=q1(t), by (25) and (26), we obtain
(27)∫01‍qi(t)dt=∫01‍q1(t+i-1N)dt=∫01‍q1(t)dt=0.
Combined with (17) and (27), we have ∫01‍(q(t)-qi(t))dt=0. Hence, by (2) of Lemma 11, we can get
(28)1N∫01‍∑i≠i0‍[12|q˙-q˙i|2+N|q-qi|]dt≥(N-1N)32(2π)2/3N2/3.
For the other term of f, using the expression for the orbits q1,…,qN as in (1), one has
(29)-12N∫01‍∑i=1N‍|q˙i|2dt=-12(2π)2r2.
Therefore, it follows from (24), (28), and (29) that
(30)infq∈∂Λ-f(q)≥32(4π)2/3N-1/3+(N-1N)32(2π)2/3N2/3-12(2π)2r2≜d0.Lemma 14.
For∂Λ+={q∈W1,2(R/Z,R2)∣q(t+1/N)=R(N)q(t),∃1≤i~0≤N,t~0∈[0,1]s.t.qi~0(t~0)=q(t~0)}, we have
(31)infq∈∂Λ+f(q)≥32(2π)2/3N1/3+(N-1N)32(2π)2/3N2/3-12(2π)2r2≜d1.Proof.
Similar to the proof of Lemma13, Lemma 14 holds.Proof of Theorem2.
In order to get Theorem2, we are going to find a test loop q~∈Λ- such that f(q~)=d2. Then the minimizer of f on Λ¯- must be a noncollision solution if d2<d0.
Leta>0, b>0, θ∈[0,2π) and
(32)q~-qN=(acos(-2πt+θ),bsin(-2πt+θ))T.
Hence,
(33)q~-q1=(rcos2πt-rcos(2πt+2πN)q~-q1=+acos(-2πt+θ),rsin2πtq~-q1=-rsin(2πt+2πN)+bsin(-2πt+θ))T,q~-q1=⋮(34)q~-qi=(rcos2πt-rcos(2πt+iN2π)q~-qi=+acos(-2πt+θ),rsin2πtq~-qi=-rsin(2πt+iN2π)+bsin(-2πt+θ))T,q~-qi=⋮(35)q~-qN-1=(rcos2πt-rcos(2πt+N-1N2π)+acos(-2πt+θ),rsin2πt-rsin(2πt+N-1N2π)+bsin(-2πt+θ))T.
It is easy to see thatq~∈Λ- and
(36)|q~˙-q˙1|2=(2π)2{a2+b22-a2-b22cos(4πt-2θ)+r2(2-2cos2πN)-(a+b)r×[cos(4πt-θ)-cos(4πt+2πN-θ)]+(a-b)r[cosθ-cos(2πN+θ)]a2+b22},(37)|q~-q1|={a2+b22+a2-b22cos(4πt-2θ)+r2(2-2cos2πN)+(a+b)r×[cos(4πt-θ)-cos(4πt+2πN-θ)]+(a-b)r[cosθ-cos(2πN+θ)]a2+b22}1/2,⋮(38)|q~˙-q˙i|2=(2π)2{a2+b22-a2-b22cos(4πt-2θ)+r2(2-2cosiN2π)-(a+b)r×[cos(4πt-θ)-cos(4πt+iN2π-θ)]+(a-b)r[cosθ-cos(iN2π+θ)]a2+b22},(39)|q~-qi|={a2+b22+a2-b22cos(4πt-2θ)+r2(2-2cosiN2π)+(a+b)r×[cos(4πt-θ)-cos(4πt+iN2π-θ)]+(a-b)r[cosθ-cos(iN2π+θ)]a2+b22}1/2,⋮(40)|q~˙-q˙N|2=(2π)2(a2+b22-a2-b22cos(4πt-2θ)),(41)|q~-qN|=a2+b22+a2-b22cos(4πt-2θ),(42)|q˙1|2=⋯=|q˙N|2=(2π)2r2.
Therefore by (36)–(42), we get
(43)f(q~)=1N∫01‍∑i=1N‍[12|q~˙-q˙i|2+N|q~-qi|]dt-12N∫01‍∑i=1N‍|q˙i|2dt=(a2+b2)π2+(2π)2r2+2π2(a-b)r[cosθ-1N∑i=1N‍cos(iN2π+θ)]+∑i=1N‍∫01‍{a2+b22+a2-b22cos(4πt-2θ)+r2(2-2cosiN2π)+(a+b)r×[cos(4πt-θ)-cos(4πt+iN2π-θ)]+(a-b)r[cosθ-cos(iN2π+θ)]a2+b22}-1/2dt-12(2π)2r2=2π2(a2+b22+r2)+2π2(a-b)r[cosθ-1N∑i=1N‍cos(iN2π+θ)]+∑i=1N‍∫01‍{a2+b22+a2-b22cos(4πt-2θ)+r2(2-2cosiN2π)+(a+b)r×[cos(4πt-θ)-cos(4πt+iN2π-θ)]+(a-b)r[cosθ-cos(iN2π+θ)]a2+b22}-1/2dt=d2(a,b,θ).
In order to estimated2, we have computed the numerical values of d2=f(q) over some selected test loops. The computation of the integral that appears in (43) has been done using the function {quad} of Mathematica 7.1 with an error less than 10-6. The results of the numerical explorations are given in Table 1.
For the parametersa,b, and θ given in Table 1, we all have d2<d0. For 4≤N≤1000, we do not find suitable parameters a∈[0.01,2r), b∈[0.01,2r), and  θ=π,π/2,π/3,…,π/29,π/30 such that d2<d0, where r is the radius of the first N bodies with equal masses. Hence we only consider the case N=2,3. This completes the proof of Theorem 2.Table 1
Parameters of test loops for Theoerm2.
N
a
b
θ
d
0
d
2
2
0.11
0.29
π
/
2
9.813298
9.753899
2
0.13
0.25
π
9.813298
9.753243
2
0.13
0.27
π
9.813298
9.643849
2
0.15
0.21
π
9.813298
9.790068
2
0.15
0.23
π
/
5
9.813298
9.746803
2
0.17
0.19
π
9.813298
9.751237
2
0.17
0.21
π
/
5
9.813298
9.625193
2
0.19
0.17
π
9.813298
9.784859
2
0.19
0.19
π
9.813298
9.564603
2
0.21
0.21
π
9.813298
9.279445
2
0.23
0.15
π
9.813298
9.793205
2
0.23
0.23
π
/
5
9.813298
9.109854
2
0.25
0.13
π
/
2
9.813298
9.730925
2
0.25
0.25
π
/
5
9.813298
9.038706
3
0.15
0.67
π
/
30
11.523843
11.505860
3
0.15
0.69
π
/
30
11.523843
11.444212
3
0.17
0.65
π
/
30
11.523843
11.493238
3
0.17
0.67
π
/
20
11.523843
11.452135
3
0.17
0.69
π
/
20
11.523843
11.400124
3
0.19
0.63
π
/
30
11.523843
11.519350
3
0.19
0.65
π
/
20
11.523843
11.455969
3
0.19
0.67
π
/
20
11.523843
11.386608
3
0.61
0.23
π
11.523843
11.516685
3
0.63
0.19
π
11.523843
11.489791
3
0.63
0.21
π
11.523843
11.436105
3
0.65
0.17
π
11.523843
11.461786
3
0.65
0.19
π
11.523843
11.392115
3
0.65
0.21
π
11.523843
11.349366
3
0.67
0.15
π
11.523843
11.472422
3
0.67
0.17
π
11.523843
11.383978
3
0.67
0.19
π
11.523843
11.324970
3
0.69
0.13
π
11.523843
11.522980
3
0.69
0.15
π
11.523843
11.412094
3
0.69
0.17
π
11.523843
11.334189Proof of Theorem4.
To get Theorem4, we are going to find a test loop q¯∈Λ+ such that f(q¯)=d3. Then the minimizer of f on Λ¯+ must be a noncollision solution if d3<d1.
Leta>0,  θ∈[0,2π), and
(44)q¯-qN=ae-12πte-1θ.
Hence,
(45)q¯-q1=qN+ae-12πte-1θ-q1=re-12πt(1-e-1(2π/N))+ae-12πte-1θ,⋮(46)q¯-qi=qN+ae-12πte-1θ-qi=re-12πt(1-e-1(i/N)2π)+ae-12πte-1θ,⋮(47)q¯-qN-1=qN+ae-12πte-1θ-qN-1=re-12πt(1-e-1((N-1)/N)2π)+ae-12πte-1θ.
It is easy to see that q¯∈Λ+ and
(48)|q¯˙-q˙1|2=(2π)2{r2(2-2cos2πN)+a2+2ar[cosθ-cos(θ-2πN)]},(49)|q¯-q1|={r2(2-2cos2πN)+a2+2ar[cosθ-cos(θ-2πN)]}1/2,⋮(50)|q¯˙-q˙i|2=(2π)2{r2(2-2cosiN2π)+a2+2ar[cosθ-cos(θ-iN2π)]},(51)|q¯-qi|={r2(2-2cosiN2π)+a2+2ar[cosθ-cos(θ-iN2π)]}1/2,⋮(52)|q¯˙-q˙N|2=(2π)2a2,|q¯-qN|=a,(53)|q˙1|2=⋯=|q˙N|2=(2π)2r2.
Therefore by (48)–(53), we get
(54)f(q¯)=1N∫01‍∑i=1N‍[12|q¯˙-q˙i|2+N|q¯-qi|]dt-12N∫01‍∑i=1N‍|q˙i|2dt=2π2a2+(2π)2r2+(2π)2[arcosθ-arN∑i=1N‍cos(θ-iN2π)]+a-1+∑i=1N-1‍{r2(2-2cosiN2π)+a2+2ar[cosθ-cos(θ-iN2π)]}-1/2-12(2π)2r2=2π2(a2+r2)+(2π)2[arcosθ-arN∑i=1N‍cos(θ-iN2π)]+a-1+∑i=1N-1‍{r2(2-2cosiN2π)+a2+2ar[cosθ-cos(θ-iN2π)]}-1/2=d3(a,θ).
In order to estimated3, we have computed the numerical values of d3=f(q) over some selected test loops. The results of the numerical explorations show that by choosing some a∈[0.11,12.50] and θ=π, one has d3<d1 for 2≤N≤32402. From the computational data, let θ=π,π/2,π/3,…,π/19,π/20, and we find that d3<d1 holds for 2≤N≤19 with some a∈[0.11,0.90]. But for 20≤N≤32402, no matter how we choose the value of a∈[0.11,12.50], we obtain d3<d1 only if θ=π. For N=32403, the computational data shows that d3≫d1 for all a∈[0.11,10.00] and θ=π,π/2,π/3,…,π/99,π/100. Then combined with Figure 1, it is easy to see that a.e.d3>d1 for all a∈[0.11,12.50] and θ=π,π/2,π/3,…,π/99,π/100. The same holds for N=32404 from Figure 2 and the computational data. Moreover, we can get the same conclusions for 32405≤N≤100000. Hence, we only consider the case 2≤N≤32402. This completes the proof of Theorem 4.Figure 1
N
=
32403.Figure 2
N
=
32404.



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*Source: 173639-2013-05-13.xml*