Nonplanar Periodic Solutions for Spatial Restricted 3-Body and 4-Body Problems

In this paper, we study the existence of non-planar periodic solutions for the following spatial restricted 3-body and 4-body problems: for $N=2 or 3$, given any masses $m_{1},...,m_{N}$, the mass points of $m_{1},...,m_{N}$ move on the $N$ circular obits centered at the center of masses, the sufficiently small mass moves on the perpendicular axis passing the center of masses. Using variational minimizing methods, we establish the existence of the minimizers of the Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, we prove these minimizers are non-planar periodic solutions by using the Jacobi's Necessary Condition for local minimizers.


Introduction and Main Results
In this paper, we study the spatial circular restricted 3-body and 4-body problems. For N = 2 or 3, suppose points of positive masses m 1 , · · · , m N move in the plane of their circular orbits q 1 (t), · · · , q N (t) with the radius r 1 , · · · , r N > 0 and the center of masses is at the origin; suppose the sufficiently small mass point does not influence the motion of m 1 , · · · , m N , and moves on the vertical axis of the moving plane for the given masses m 1 , · · · , m N , here the vertical axis passes through the center of masses.
It is known that q 1 (t), · · · , q N (t)(N = 2 or 3) satisfy the Newtonian equations: The orbit q(t) = (0, 0, z(t)) ∈ R 3 for sufficiently small mass is governed by the gravitational forces of m 1 , · · · , m N (N = 2 or 3) and therefore it satisfies the following equation For N ≥ 2, there are many papers concerned with the restricted N-body problem, see [3,4,6,[8][9][10] and the references therein. In [8], Sitnikov considered the following model: two mass points of equal mass m 1 = m 2 = 1 2 move in the plane of their elliptic orbits and the center of masses is at rest, the third mass point which does not influence the motion of the first two moves on the line perpendicular to the plane containing the first two mass points and goes through the center of mass, and he used geometrical methods to prove the existence of the oscillatory parabolic orbit of where r(t) = r(t+2π) > 0 is the distance from the center of mass to one of the first two mass points.
McGehee [6] used the stable and unstable manifolds to study the homoclinic orbits (parabolic orbits) of (1.4). In [4], Mathlouthi studied the periodic solutions for the spatial circular restricted 3-body problems by minimax variational methods. Recently, Li, Zhang and Zhao [3] used variational minimizing methods to study spatial circular restricted N+1-body problem with a zero mass moving on the vertical axis of the moving plane for N equal mass.
Motivated by [3], we use the Jacobi's Necessary Condition for local minimizers to further study the spatial circular restricted 3-body and 4-body problems with a sufficiently small mass moving on the perpendicular axis of the circular orbits plane for any given masses m 1 , · · · , m N (N = 2 or 3). Define The inner product and the norm of The functional corresponding to the equation (1. 3) is where and Our main results are the following:

Preliminaries
In this section, we will list some basic Lemmas and inequality for proving our Theorems 1.1 and 1.2.
Lemma 2.1(Palais's Symmetry Principle( [7])) Let σ be an orthogonal representation of a finite or compact group G, H be a real Hilbert space, f : Then the critical point of f in F is also a critical point of f in H. Lemma 2.2(Tonelli [1]) Let X be a reflexive Banach space, S be a weakly closed subset of X, f : S → R ∪ +∞. If f ≡ +∞ is weakly lower semi-continuous and coercive(f (x) → +∞ as x → +∞), then f attains its infimum on S.
Hence by the definitions of f (q), it is easy to see that f is C 1 and coercive on Λ i (i = 1, 2). In order to get Lemma 2.4, we only need to prove that f is weakly lower semi-continuous on Λ i (i = 1, 2). In fact, for ∀z n ∈ Λ i , if z n ⇀ z weakly, by compact embedding theorem, we have the uniformly convergence: max It is well-known that the norm and its square are weakly lower semi-continuous. Therefore, combined with (2.4), one has lim inf that is, f is weakly lower semi-continuous on Λ i (i = 1, 2). By lemma 2.2, we can get that f (q) in (1.7) attains its infimum onΛ i = Λ i (i = 1, 2).
, R)|y(a) = A, y(b) = B} and if F y ′ y ′ > 0 along this critical point, then the open interval (a, b) contains no points conjugate to a, that is, for ∀c ∈ (a, b), the following problem: Remark 2.2 It is easy to see that Lemma 2.5 is suitable for the fixed end problem. In this paper, we consider the periodic solutions of (1.3) on Λ i = Λ i (i = 1, 2), hence we need to establish a similar conclusion as Lemma 2.5 for the periodic boundary problem.

Proof of Theorem 1.1
In this section, we consider the spatial circular restricted 3-body problem with a sufficiently small mass moving on the vertical axis of the moving plane for arbitrary given positive masses m 1 , m 2 . Suppose the planar circular orbits are here the radius r 1 , r 2 are positive constants depending on m i (i = 1, 2) and T (see Lemma 3.1). We also assume that m 1 q 1 (t) + m 2 q 2 (t) = 0. (3. 2) The functional corresponding to the equation (1.3) is

Proof. Substituting (3.1) into (3.2), it is easy to get
which implies (3.7) Hence by (3.4), one has Then the second variation of (3.3) in the neighborhood of z = 0 is given by (3.11) The Euler equation of (3.9) is called the Jacobi equation of the original functional (3.3), which is Next, we study the solution of (3.13) with initial values h(0) = 0, h ′ (0) = 1. It is easy to get (3.14) It follows from (3.7) and (3.8)that Hence ]. Since Therefore Case 1: Minimizing f (q) onΛ 1 = Λ 1 .
Case 2: Minimizing f (q) onΛ 2 = Λ 2 . (3.23) andh is a nonzero solution of (3.12). Notice that we can extendh periodically when we take T as the period, soh ∈ Λ 2 . Then by Lemma 2.6, q(t) = (0, 0, 0) is not a local minimum for f (q) on Λ 2 . Hence the minimizers of f (q) on Λ 2 are not always at the center of masses, they must oscillate periodically on the vertical axis, that is, the minimizers are not always co-planar, therefore, we get the non-planar periodic solutions.
Combined with Lemma 2.4, the proof is completed.

Proof of Theorem 1.2
In this section, we consider the spatial circular restricted 4-body problem with a sufficiently small mass moving on the vertical axis of the moving plane for arbitrary positive masses m 1 , m 2 , m 3 . Suppose there exists θ 1 , θ 2 , θ 3 ∈ (0, 2π) such that the planar circular orbits are here the radius r 1 , r 2 , r 3 are positive constants depending on m i (i = 1, 2, 3) and T (see Lemma 4.2). We also assume that m 1 q 1 (t) + m 2 q 2 (t) + m 3 q 3 (t) = 0 (4.2) and where the constant l > 0 depends on m i (i = 1, 2, 3) and T (see Lemma 4.1). The functional corresponding to the equation (1.3) is  Proof. It follows from (1.1) and (1.2) thaẗ (4.5) Then the second variation of (4.4) in the neighborhood of z = 0 is given by Case 1: Minimizing f (q) onΛ 1 = Λ 1 . (4.25) It is easy to check thath(t) ∈ C 2 ([0, T ]\{c, T 2 , T 2 + c}, R) ∩ W 1,2 (R, R),h(t + T 2 ) = −h(t),h(0) = h(0) = 0,h(c) = h(c) = 0 andh is a nonzero solution of (4.16). Notice that we can extendh periodically when we take T as the period, soh ∈ Λ 1 . Then by Lemma 2.6, q(t) = (0, 0, 0) is not a local minimum for f (q) on Λ 1 . Hence the minimizers of f (q) on Λ 1 are not always at the center of masses, they must oscillate periodically on the vertical axis, that is, the minimizers are not always co-planar, therefore, we get the non-planar periodic solutions.
Combined with Lemma 2.4, the proof is completed.