Families of Periodic Orbits for the Spatial Isosceles 3-body Problem

We study the families of periodic orbits of the spatial isosceles 3-body problem (for small enough values of the mass lying on the symmetry axis) coming via the analytic continuation method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic orbits of the reduced isosceles problem generate invariant two-dimensional tori of the nonreduced problem, the analytic continuation of periodic orbits of the (reduced) circular Sitnikov problem at this level becomes the continuation of invariant two-dimensional tori from the circular Sitnikov problem to the nonreduced isosceles problem, each one filled with periodic or quasi-periodic orbits. These tori are not KAM tori but just isotropic, since we are dealing with a three-degrees-of-freedom system. The continuation of periodic orbits is done in two different ways, the first going directly from the reduced circular Sitnikov problem to the reduced isosceles problem, and the second one using two steps: first we continue the periodic orbits from the reduced circular Sitnikov problem to the reduced elliptic Sitnikov problem, and then we continue those periodic orbits of the reduced elliptic Sitnikov problem to the reduced isosceles problem. The continuation in one or two steps produces different results. This work is merely analytic and uses the variational equations in order to apply Poincaré's continuation method. 1. Introduction. We consider a special case of the spatial 3-body problem, the spatial isosceles 3-body problem, or simply the isosceles problem. This problem consists of describing the motion of two equally massive bodies, m 1 = m 2 = 1/2, having initial conditions and velocities symmetric with respect to a straight line which passes through their center of mass, and a third body, with mass m 3 = µ, having initial position and velocity on this straight line. This problem is called the isosceles problem because the three bodies form an isosceles triangle at any time, eventually degenerated to a segment. The most interesting application of the spatial isosceles 3-body problem was given by Xia in [25]. He used two spatial isosceles 3-body problems to prove that five bodies can escape to infinity in a finite time without collision. Other works on the spatial isosceles 3-body problem are [16] and the references therein. If in the spatial isosceles 3-body problem the initial positions and velocities of the three bodies …


Introduction.
We consider a special case of the spatial 3-body problem, the spatial isosceles 3-body problem, or simply the isosceles problem.This problem consists of describing the motion of two equally massive bodies, m 1 = m 2 = 1/2, having initial conditions and velocities symmetric with respect to a straight line which passes through their center of mass, and a third body, with mass m 3 = µ, having initial position and velocity on this straight line.This problem is called the isosceles problem because the three bodies form an isosceles triangle at any time, eventually degenerated to a segment.
The most interesting application of the spatial isosceles 3-body problem was given by Xia in [25].He used two spatial isosceles 3-body problems to prove that five bodies can escape to infinity in a finite time without collision.Other works on the spatial isosceles 3-body problem are [16] and the references therein.If in the spatial isosceles 3-body problem the initial positions and velocities of the three bodies are contained in a plane, then the motion remains always in this plane, and we have the so-called planar isosceles 3-body problem.There are several papers about the planar isosceles 3-body problem, for instance, [9], [17], etc.
When the third body of the isosceles 3-body problem has infinitesimal mass (i.e., µ = 0) then we obtain the restricted isosceles problems.Depending on the motion * Received by the editors May 17, 2002; accepted for publication (in revised form) May 30, 2003; published electronically January 30, 2004.This work was partially supported by MCYT grants BFM 2002-04236-C02-02 and by CIRIT grant SGR 2001 00173.
‡ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (jllibre@mat.uab.es).1311 Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpTheorem A. Let γ be a periodic orbit of the reduced circular Sitnikov problem with period T > π/ √ 2, and let f (e) = (1 − e 2 ) 3/2 .Then γ can be continued to the following families of periodic orbits of the reduced isosceles problem with angular momentum c = 1/4 and µ > 0 sufficiently small: 1. Case T = 2πω with ω > 1/(2 √ 2) an irrational number.(a) γ can be continued directly to one 2-parameter family (on µ and τ ) of doubly symmetric periodic orbits with period τ sufficiently close to T .2. Case T = 2πp/q for some p, q ∈ N coprime with p > q/(2 √ 2).(a) p odd: i. γ can be continued directly to one 2-parameter family (on µ and τ ) of doubly symmetric periodic orbits with period τ sufficiently close to T .ii. γ can be continued by two steps to two 2-parameter families (on µ and e) of r-symmetric periodic orbits with period qT f (e) where e > 0 is sufficiently small.iii.γ can be continued by two steps to two 2-parameter families (on µ and e) of t-symmetric periodic orbits with period qT f (e) where e > 0 is sufficiently small.(b) p even and q = 1: i. γ can be continued directly to one 2-parameter family (on µ and τ ) of doubly symmetric periodic orbits with period τ sufficiently close to T .ii. γ can be continued by two steps to two 2-parameter families (on µ and e) of doubly symmetric periodic orbits of period qT f (e) where e > 0 is sufficiently small.(c) p even and q = 1: i. γ can be continued by two steps to two 2-parameter families (on µ and e) of doubly symmetric periodic orbits of period qT f (e) where e > 0 is sufficiently small.Using direct continuation we can continue all periodic orbits of the reduced circu-of the isosceles problem in appropriate cylindrical coordinates; these coordinates will allow us to define the reduced isosceles problem in section 3.In section 4 we give the relationships between the orbits of the reduced isosceles problem and the isosceles problem.In particular, we see that if ϕ is an orbit for the reduced isosceles problem, then ϕ × S 1 is an invariant manifold for the isosceles problem (for more details see Theorem 4.1).In section 5 we analyze the symmetries of the reduced isosceles problem.In section 6 we define the restricted isosceles problems and the reduced restricted isosceles problems.In this work, we will consider only the circular and elliptic restricted isosceles problems, which are treated in sections 7 and 8, respectively.In particular, we are interested in the invariant two-dimensional tori of these problems that come from periodic orbits of the corresponding reduced problems.In section 7.1, we summarize the basic properties given in [8] of the periodic solutions of the circular Sitnikov problem.In section 8.1 we summarize the basic properties of the periodic solutions of the elliptic Sitnikov problem and give the basic results on continuation of periodic solutions from the circular Sitnikov problem (e = 0) to the elliptic Sitnikov problem for e > 0 sufficiently small.These results have also been extracted from [8].In section 9 we analyze the variational equations of the reduced circular and elliptic Sitnikov problem and explicitly give the solution of the variational equations of the Kepler problem along a circular or elliptic periodic solution and the solution of the variational equations of the circular Sitnikov problem.In section 10 we analyze the direct continuation of periodic solutions from the reduced circular Sitnikov problem to the isosceles problem for µ > 0 sufficiently small; in particular, we prove statements 1(a), 2(a)i, and 2(b)i of Theorem A (see Theorem 10.1).In section 11 we analyze the continuation of the symmetric periodic solutions of the reduced elliptic Sitnikov problem that we give in section 8 to the isosceles problem for µ > 0 sufficiently small.The continuation by two steps from the reduced circular Sitnikov problem to the reduced isosceles problem is analyzed in section 12; in particular, we prove the remaining statements of Theorem A (see Theorem 12.8).In section 13 we summarize the basic results on continuation of invariant two-dimensional tori from the circular restricted isosceles problem to the isosceles problem for µ > 0 small.

Coordinates and equations of motion of the isosceles problem.
Let P 1 and P 2 be two particles, with equal masses m 1 = m 2 , having initial positions and velocities symmetric with respect to a straight line that passes through their center of mass.Let P 3 be a third particle, with mass m 3 , having initial position and velocity on this straight line.The spatial isosceles 3-body problem, or simply the isosceles problem in this work, consists of describing the motion of these three particles under their mutual Newtonian gravitational attraction.We note that the solutions of the isosceles problem are in fact solutions of the general spatial 3-body problem.
In order to develop our analysis we will use the cylindrical coordinates (r, z, θ) ∈ R + × R × S 1 introduced as follows.Here R + denotes the open interval (0, ∞).First we put the origin 0 of the coordinate system at the center of mass of m 1 , m 2 , and m 3 , which implies taking Z 2 = −m 3 Z 1 .Then we define a new variable z = Z 1 − Z 2 ∈ R which denotes the distance between the third particle P 3 and the orthogonal plane to the Z-axis that contains the particles P 1 and P 2 with the convenient sign (positive Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php . Finally we consider polar coordinates, (r, θ) ∈ R + × S 1 , in the above orthogonal plane by taking X = r cos θ and Y = r sin θ.
We choose the unit of mass in such a way that m 1 = m 2 = 1/2 and m 3 = µ, and the unit of length is chosen so that the gravitational constant is one.Then the kinetic energy and the potential energy in the coordinate system (r, ṙ, z, ż, θ, θ) are given, respectively, by Therefore the Lagrangian equations of motion for the isosceles problem are We note that the third equation of system (2.1) can be integrated directly, obtaining the first integral of the angular momentum Of course, system (2.1) also has the first integral given by the energy H = T + U .

The reduced isosceles problem.
To avoid singular situations, throughout this work we consider only solutions of system (2.1) having nonzero angular momentum (i.e., in particular, we do not consider solutions with collision between the masses, either triple or double).We note that under this assumption it is sufficient to consider solutions of (2.1) having a fixed value of the angular momentum C = c for some c = 0, Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpbecause the phase portrait of the isosceles problem on each angular momentum level c = 0 is the same as that shown in the following proposition.
Proposition 3.1.Let (r(t), ṙ(t), z(t), ż(t), θ(t), θ(t)) be a solution of the isosceles problem (2.1) with angular momentum C = c for some c = 0.If we take α 1/2 = c/c = 0, then is a solution of (2.1) with angular momentum c.Proof.It is easy to see that system (2.1) is invariant under the transformation Thus ϕ(t) is a solution of (2.1).Moreover the angular momentum of ϕ(t) is given by Then ϕ(t) is a solution of (2.1) with angular momentum c.
Assuming that the value of the angular momentum is fixed at C = c for some c = 0, we can reduce by two units the dimension of the phase space.Indeed, the variable θ does not appear explicitly in system (2.1); moreover from (2.2), θ = c/r 2 , and thus we need to consider only the first two equations of (2.1) with θ replaced by c/r 2 .That is, we need to consider only the reduced isosceles problem
Fixing a value of c = 0, the union of the orbits γ ϕ,θ0,c = {γ ϕ,θ0,c (t By the qualitative theory of differential equations we know that the orbits of the reduced isosceles problem (3.1) can be either equilibrium points, periodic orbits, or Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phporbits diffeomorphic to R. Thus if ϕ is an equilibrium point, then the corresponding relative set is diffeomorphic to a circle S 1 (a relative periodic orbit).If ϕ is a periodic orbit (i.e., a closed curve diffeomorphic to S 1 ), then the corresponding relative set is diffeomorphic to a two-dimensional torus S 1 × S 1 (a relative torus).This relative torus can be filled with either periodic or quasi-periodic orbits (in this last case the orbits are dense on the torus).We note that these kinds of tori are not KAM tori (see, for instance, [1]), because they are two-dimensional tori of a problem with three degrees of freedom, and the KAM tori of such a system have dimension 3. Finally if ϕ is neither an equilibrium point nor a periodic orbit, then the corresponding relative set is diffeomorphic to a cylinder R × S 1 .In particular, we have the following result.

A two-dimensional torus
this torus is formed by the union of (a) periodic orbits of period mT if

Symmetries.
It is easy to check that the equations of motion of the reduced isosceles problem (3.1) are invariant under the symmetry (t, r, ṙ, z, ż) −→ (−t, r, − ṙ, −z, ż).(5.1)This means that if ϕ(t) = (r(t), ṙ(t), z(t), ż(t)) is a solution of system (3.1), then also ψ(t) = (r(−t), − ṙ(−t), −z(−t), ż(−t)) is a solution.We note that in the configuration space {(r, z) ∈ R + × R} this symmetry corresponds to a symmetry with respect to the r-axis, so in what follows it will be denoted by the r-symmetry.On the other hand, in the configuration space {(r, z, θ) ∈ R + × R × S 1 } the r-symmetry would correspond to a symmetry with respect to the plane defined by the motion of the particles P 1 and P 2 .
This symmetry can be used, in a standard way, to find periodic solutions as follows.Suppose that ϕ(t) crosses orthogonally the r-axis at a time t = 0; that is, z(0) = 0 and ṙ(0) = 0. Using symmetry (5.1) we have that the two solutions ϕ(t) and ψ(t) coincide at t = 0; then by the theorem of uniqueness of solutions of an ordinary differential equation they must be the same.If there is another time such that the solution ϕ(t) crosses the r-axis orthogonally, then by symmetry (5.1) the orbit of ϕ(t) must be closed, and ϕ(t) is called an r-symmetric periodic solution.
Since system (3.1) is autonomous, the origin of time can be chosen arbitrarily.Thus, if γ(t) is a solution of (3.1) that crosses the r-axis in a point p at t = t 0 , then ϕ(t) = γ(t + t 0 ) is a solution of (3.1) that crosses the r-axis in the point p at t = 0. Therefore we have proved the following well-known result.
Equations (3.1) are also invariant under the symmetry (t, r, ṙ, z, ż) −→ (−t, r, − ṙ, z, − ż), (5.2) i.e., the time reversibility symmetry, which will be denoted in what follows by the tsymmetry.As in the r-symmetry we can introduce the notion of t-symmetric periodic solutions, which are characterized as follows.
We note that there could be periodic solutions of (3.1) that are simultaneously r-and t-symmetric.These periodic solutions will be called doubly symmetric periodic solutions (see, for instance, [12] for more information about doubly symmetric periodic orbits) and are characterized by the following result. Proposition

Restricted isosceles problems.
To obtain the restricted isosceles problems we assume that the value of the mass m 3 is infinitesimally small (i.e., µ = 0).Then the equations of motion of the restricted isosceles problem become Notice that the first and the third equations of (6.1) do not depend on z; moreover they are the equations of motion of a 2-body problem in polar coordinates.This means that the particles P 1 and P 2 (the primaries) move on the plane z = 0 describing a solution of this 2-body problem.Moreover the particle P 3 that lies on the straight line orthogonal to the plane containing P 1 and P 2 that passes through their center of mass moves under the gravitational attraction of the previous two but does not influence their motion.Thus, for every solution (r(t), θ(t)) of that 2-body problem, system (6.1)defines a different restricted isosceles problem; it can be a circular, elliptic, parabolic, hyperbolic, elliptic collision, parabolic collision, or hyperbolic collision restricted isosceles problem depending on the nature of the solution (r(t), θ(t)).
As in the isosceles problem (2.1) if we assume that the value of the angular momentum is fixed at C = c for some c = 0, then we can reduce the dimension of the phase space by two, obtaining the reduced restricted isosceles problem In this work we are interested only in the periodic solutions of system (6.2) for c = 0. So, we will consider only the reduced circular and elliptic restricted isosceles Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpproblems, which we will call reduced circular Sitnikov problem and reduced elliptic Sitnikov problem, respectively.7. On the circular restricted isosceles problem.Without loss of generality we can assume that the primaries describe a circular orbit of radius 1/2 (or, equivalently, a circular orbit of period 2π).This corresponds to fixing the value of the angular momentum to c = 1/4.Then the equation of motion for the infinitesimal mass becomes which is the equation of the known circular Sitnikov problem.
Since we have taken r(t) = 1/2, ṙ(t) = 0, it's clear that ϕ(t) = (r(t), ṙ(t), z(t), ż(t)) is a periodic solution of the reduced circular Sitnikov problem with period T if and only if (z(t), ż(t)) is a periodic solution of the circular Sitnikov problem (7.1) with period T .So, we start summarizing the basic results about periodic solutions of the circular Sitnikov problem (7.1) that are needed for the development of this work.Then we will analyze the periodic solutions of the reduced circular Sitnikov problem and their relationship with the corresponding solutions of the circular restricted isosceles problem.

Periodic solutions of the circular Sitnikov problem. Equation (7.1) defines an integrable Hamiltonian system of one degree of freedom with Hamiltonian
, where v = ż.The orbits for the circular Sitnikov problem in the energy level h are described by the curve H = h, where h varies in [−2, ∞).The circular Sitnikov problem has been studied by several authors.In 1907 Pavanini [19] expressed its solutions by means of Weierstrassian elliptic functions.Four years later MacMillan [14] expressed the solutions in terms of Jacobian elliptic functions (a detailed description of this work can be found in Stumpff [22]).Some other analytical expressions for the solutions of this problem can be found, for instance, in [23], [2], and [24].In particular, in this paper we will use the analytical expressions of the solutions of the circular Sitnikov problem for h > −2 that appear in [2], which are given in terms of Jacobian elliptic functions.A detailed description of all Jacobian elliptic functions to be used in this paper can be found in [4] and [8].
We remark that the knowledge of an analytic expression for the solutions of the circular Sitnikov problem plays a key role in our analysis, because it allows us to prove our results analytically.
Using the analytic expression for the solutions of the circular Sitnikov problem given in Theorem A of [2], we see that the periodic solutions of that problem can be written as follows (see [8] for more details).
Lemma 7.1.The periodic solutions of the circular Sitnikov problem have energy −2 < h < 0 and can be written as where k = √ 2 + h/2 and ν is the function of t defined implicitly by Here C is an integration constant whose value depends on the initial conditions of the periodic solution (z(t), ż(t)).
Since sn ν and cn ν are periodic functions of period 4K and dn ν is a periodic function of period 2K (see formulas 122 in [4]), from (7.3) we see that the period in the new time ν is 4K, where K = K(k) is the complete elliptic integral of the first kind and k = √ 2 + h/2.Moreover the period in the real time t is given by for more details see Theorem 2.3 in [8].
We note that (7.1) is autonomous, so the origin of time can be chosen arbitrarily.In particular, in this paper we are interested only in periodic solutions (z(t), ż(t)) having initial conditions either z(0) = 0 or ż(0) = 0.The following lemma, taken from [8], gives the values of the integration constant C for those initial conditions.Lemma 7.2.Let T be the period of the periodic solution (z(t), ż(t)) given in (7.4).
1.If (z(t), ż(t)) has initial conditions z(0) = 0 and ż(0 In order to simplify computations we will usually work with the new time ν instead of the real time t, but always keeping in mind that ν is a function of t via Lemma 7.2.The two following lemmas taken also from [8] give some relationships between the real time t and the new time ν that will be useful later on. Lemma 7.3.Let T be the period of the periodic solution (z(t), ż(t)).1. ν(t + qT ) = ν(t) + q4K for all t ∈ R and for all q ∈ N.
The following result gives the properties of the function T = T (h).Theorem 7.5.The period T satisfies Proof.See the proof of Theorem C in [2].Theorem 7.5 assures the existence of periodic orbits of the circular Sitnikov problem with period T = T (h) for all T > π/ √ 2. In fact, since T = T (h) is an injective function there is a one-to-one correspondence between h ∈ (−2, 0) and T ∈ (π/ √ 2, ∞), so we can characterize the periodic orbits either by the period or by the energy.

Periodic solutions of the reduced circular Sitnikov problem.
Notice that equations (7.2) are invariant under symmetries (5.1) and (5.2).These symmetries can be used to obtain symmetric periodic solutions for the reduced circular Sitnikov problem.It is not difficult to prove the next result.
Proposition 7.6.All periodic orbits of the reduced circular Sitnikov problem are doubly symmetric periodic orbits.
We note that the periodic solutions of the reduced circular Sitnikov problem are periodic solutions for the infinitesimal mass, but in general they are not periodic solutions involving the three masses; that is, they are not periodic solutions of the circular restricted isosceles problem.Since the primaries describe a circular solution of a 2-body problem with period 2π, the only periodic orbits of the circular Sitnikov problem that give periodic orbits involving the three masses are the ones that have a period commensurable with 2π; that is, T = T (h) = 2πp/q for some p, q ∈ N coprime.In this case the period of the corresponding orbit involving the three masses is τ = 2πp = qT (h).That is, during a period τ , the primaries have completed p revolutions and the infinitesimal mass has completed q revolutions.

Invariant tori of the circular restricted isosceles problem.
From section 7.2, we have the following result, which can be obtained easily from Theorem 4.1.
Proposition 7.7.Let {(z h (t), żh (t)) : t ∈ R} be a periodic orbit of the circular Sitnikov problem with energy h for some h ∈ (−2, 0); and let ϕ h = {ϕ h (t) = (r(t) = 1/2, ṙ(t) = 0, z h (t), żh (t)) : t ∈ R} be its corresponding orbit of the reduced circular Sitnikov problem.Then the relative set of the circular restricted isosceles problem associated to the orbit ϕ h is diffeomorphic to a two-dimensional torus S 1 ×S 1 .Moreover, this relative torus is formed by the union of 1. periodic orbits of period qT if T = T (h) = 2πp/q for some p, q ∈ N coprime and p > q/(2 8. On the elliptic restricted isosceles problem.We assume that the primaries are describing an elliptic orbit of the 2-body problem with period 2π and eccentricity e.This corresponds to fixing the value of the angular momentum to c = c e = √ 1 − e 2 /4.Then, choosing conveniently the origin of time, a solution of the reduced elliptic Sitnikov problem is a solution Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpwith initial conditions r(0) = (1 ± e)/2, ṙ(0) = 0, z(0) = z 0 , ż(0) = ż0 for some z 0 , ż0 ∈ R.
Since r(t) is a 2π-periodic function, the periodic solutions of the reduced elliptic Sitnikov problem must have period that is a multiple of 2π.Moreover ϕ(t) = (r(t), ṙ(t), z(t), ż(t)) is a periodic solution of the reduced elliptic Sitnikov problem with period T = 2kπ for some k ∈ N if and only if (z(t), ż(t)) is a periodic solution with period T = 2kπ of the elliptic Sitnikov problem It is clear that equations (8.1) are invariant under symmetries (5.1) and (5.2).These symmetries can be used to obtain symmetric periodic solutions for the reduced elliptic Sitnikov problem.We remark that symmetries (5.1) and (5.2) for the reduced elliptic Sitnikov problem correspond to the r-and the t-symmetry of the elliptic Sitnikov problem defined in [7] and [8].

Periodic solutions of the reduced elliptic Sitnikov problem.
In section 7.2 we have seen that all periodic orbits of the reduced circular Sitnikov problem are doubly symmetric periodic orbits.This fact does not occur when we consider the reduced elliptic Sitnikov problem, as follows from the next result.
Proposition 8.1.For the reduced elliptic Sitnikov problem there exist four different types of periodic orbits: nonsymmetric periodic orbits, doubly symmetric periodic orbits, and r-and t-symmetric periodic orbits that are not doubly symmetric.
On the other hand, [8] gives initial conditions for some symmetric periodic solutions of the elliptic Sitnikov problem (or, equivalently, the reduced elliptic Sitnikov problem) with sufficiently small values of the eccentricity e > 0. These initial conditions are obtained from the analytic continuation of the known periodic solutions of the reduced circular Sitnikov problem to symmetric periodic solutions of the reduced elliptic Sitnikov problem for sufficiently small values of the eccentricity e.Later on, in section 11, the symmetric periodic solutions of the reduced elliptic Sitnikov problem given in [8] will be continued to the reduced isosceles problem for sufficiently small values of µ > 0.Here we summarize the main results of [8] about symmetric periodic orbits of the reduced elliptic Sitnikov problem.
1.This solution can be continued to two families ϕ ce (t; r 0 = (1 − e)/2, ṙ0 = 0, z 0 = z P 0 = z * 0 + O(e), ż0 = 0, µ = 0) and ϕ ce (t; r 0 = (1 + e)/2, ṙ0 = 0, z 0 = z A 0 = z * 0 +O(e), ż0 = 0, µ = 0) of t-symmetric periodic solutions of the reduced elliptic Sitnikov problem having period τ = 2πp = qT for e > 0 sufficiently small.2. If p is odd, then those t-symmetric periodic solutions are not doubly symmetric, whereas if p is even, then they are doubly symmetric.Proof.See the proof of Theorem 4.6 in [8].We note that in Theorems 8.2 and 8.3 we continue four different initial conditions of the periodic orbit of the reduced circular Sitnikov problem with period T = 2πp/q for given p, q ∈ N coprime, p > q/ ( , 0, 0) in Theorem 8.3.These four initial conditions are continued to eight families of periodic orbits of the reduced elliptic Sitnikov problem for e > 0 sufficiently small.The following theorem says how many of these eight families of periodic orbits are really different (see [8] for more details).
Theorem 8.4.The periodic solutions of the reduced circular Sitnikov problem with period T = 2πp/q, for given p, q ∈ N coprime p > q/(2 √ 2), can be continued to 1. two families of r-symmetric periodic orbits and two families of t-symmetric periodic orbits (that are not doubly symmetric) of the reduced elliptic Sitnikov problem with period τ = 2πp = qT , for e > 0 sufficiently small, when p is odd; 2. two families of doubly symmetric periodic orbits of the reduced elliptic Sitnikov problem with period τ = 2πp = qT , for e > 0 sufficiently small, when p is even.Proof.See the proof of Theorem 4.15 in [8].

Invariant tori of the elliptic restricted isosceles problem.
From Theorem 4.1(2)(a), the next result follows.
Proposition 8.5.Let ϕ = {ϕ(t) = (r(t), ṙ(t), z(t), ż(t)) : t ∈ R} be a periodic orbit of the reduced elliptic Sitnikov problem with period τ = 2πn for some n ∈ N. Then the relative set of the restricted isosceles problem associated to the orbit ϕ is diffeomorphic to a two-dimensional torus S 1 × S 1 ⊂ E ce , which is formed by periodic orbits of period τ .
We remark that the orbits of the circular restricted isosceles problem coming from periodic orbits of the reduced circular Sitnikov problem are not in general periodic orbits (see Proposition 7.7).
By means of Propositions 7.7 and 8.5, Theorem 8.4 can be extended to the restricted isosceles problem, obtaining the following result.
Theorem 8.6.Let Γ pq be the periodic two-dimensional tori of the circular restricted isosceles problem that comes from the periodic orbit of the reduced circular Sitnikov problem with period T = p2π/q, p, q ∈ N coprime and p > q/2 √ 2. Then Γ pq can be continued to two or four families (two for even p and four for odd p) of periodic two-dimensional tori of the elliptic restricted isosceles problem.9. Variational equations.The main objective of this work is to continue the known symmetric periodic orbits of the reduced circular and elliptic Sitnikov problems to symmetric periodic orbits of the reduced isosceles problem for µ > 0 sufficiently Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpsmall.Those periodic orbits will be continued by using the classical analytic continuation method of Poincaré (for details see [21] or [15]).In order to apply this method to our problem we must know the solution of the variational equations of the reduced circular and elliptic Sitnikov problems along the periodic solutions that we want to continue.In this section we will analyze those variational equations.
Let (r(t), R(t), z(t), Z(t)) be a solution of the reduced circular (e = 0) or elliptic (0 < e < 1) Sitnikov problem In particular, (r(t), R(t)) is a circular or elliptic solution of the Kepler problem (see sections 7 and 8).
The variational equations of system (9.1)along the solution curve (r(t), R(t), z(t), Z(t)) are given by the matrix differential equation with initial condition A(0) = I (the 4 × 4 identity matrix), where respectively, and If we denote q 1 = r 0 , q 2 = R 0 , q 3 = z 0 , and q 4 = Z 0 system (9.4) can be written like the linear system of differential equations, , Since equations (9.5) do not depend on ∂z/∂q i and ∂Z/∂q i , they can be solved separately.Thus, the derivatives ∂r/∂r 0 , ∂r/∂R 0 , ∂R/∂r 0 , and ∂R/∂R 0 are given by the solution of the matrix differential equation with initial condition A 1 (0) = I (the 2 × 2 identity matrix); that is, they are given by the solution of the variational equations of the Kepler problem (9.2) along the solution curve (r(t), R(t)).
On the other hand, because the first two equations of (9.1) do not depend on z and Z; consequently changes on the initial conditions z 0 and Z 0 do not affect the solution (r(t), R(t)).
By (9.6) and (9.8), the derivatives ∂z/∂z 0 , ∂z/∂Z 0 , ∂Z/∂z 0 , and ∂Z/∂Z 0 are given by the solution of the matrix differential equation with initial condition A 4 (0) = I (the 2 × 2 identity matrix); that is, they are given by the solution of the variational equations of the circular or elliptic Sitnikov problem (9.3) along the solution curve (z(t), Z(t)).
We note that we do not know an exact expression for the symmetric periodic solutions of the nonautonomous elliptic Sitnikov problem, and thus their variational equations cannot be solved explicitly.However, since the eccentricity e is sufficiently small, the solution of these variational equations may be expressed as a power series of the eccentricity e.We have computed analytically the terms of zero order in e.They are given by the variational equations of the circular Sitnikov problem.
Finally the derivatives ∂z/∂r 0 , ∂z/∂R 0 , ∂Z/∂r 0 , and ∂Z/∂R 0 are obtained by solving the nonhomogeneous linear system of differential equations that comes from replacing in (9.6) ∂r/∂q i and ∂R/∂q i , i = 1, 2, by the solutions (∂r/∂q i )(t) and (∂R/∂q i )(t) of the variational equations of the Kepler problem (9.2) along the solution curve (r(t), R(t)).If we know a fundamental matrix Φ(t) of the variational equations of the circular or elliptic Sitnikov problem (9.3) along the solution curve (z(t), Z(t)) (i.e., a fundamental matrix of the homogeneous system), then we can solve Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php the nonhomogeneous one using the method of variation of constants (see, for instance, [11, p. 81]).Thus, for i = 1, 2, we have that In order to compute the solution of the variational equations of the Kepler problem (9.2), for 0 e < 1, and of the circular Sitnikov problem (9.3) with r(t) = 1/2, we could use a theorem of Diliberto [10] on the integration of the homogeneous variational equations of a plane autonomous differential system in terms of geometric quantities along a given solution curve of the system (see also the paper of Chicone [5], where, in addition to using the Diliberto theorem to address his problem, he corrects a flaw in the theorem).But we compute here the solution of those variational equations directly using a result that appears in [8].
We note that the Kepler problem (9.2) and the circular Sitnikov problem (9.3) with r(t) = 1/2 can be written like a second order differential equation of the form ẍ = f (x).(9.10) The solution of the variational equations of (9.10) along a given nonconstant solution curve x(t) are given by the following result.

Variational equations of the Kepler problem.
We start computing a fundamental matrix of the variational equations (9.7) of the Kepler problem (9.2) for 0 e < 1 along an arbitrary elliptic solution (a circular solution if e = 0) As usual u is the eccentric anomaly which is a function of t via the Kepler's equation where M is the mean anomaly and τ is the time of pericenter passage.Later on we will give the solution of those variational equations when (r(t), R(t)) is the solution with initial conditions r(0) = (1 ± e)/2 and R = ṙ(0) = 0. Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We note that when e = 0 we cannot apply Proposition 9.1 to solve the variational equations (9.7) of the Kepler problem (9.2) along the solution curve (9.11), because r(t) = 1/2 is constant.
Proposition 9.2.When e = 0, the solution of the variational equations (9.7) of the Kepler problem (9.2) along the solution curve (9.11) is given by Proof.The proof follows easily, noting that the solution of the variational equations (9.7) when e = 0 is a matrix whose columns are the solutions of the differential equation with initial conditions ω 1 (0) = 1, ω 2 (0) = 0 and ω 1 (0) = 0, ω 2 (0) = 1, respectively.When 0 < e < 1, to solve the variational equations (9.7) of the Kepler problem (9.2) along the solution curve (9.11), we apply Proposition 9.1.Thus a fundamental matrix of those variational equations is given by where g(t) = ṙ(t) dt ṙ2 (t) .In order to simplify our computations we will work with the eccentric anomaly, u, instead of the real time, t, but keeping in mind that u is a function of t via (9.12) when it is necessary.
Replacing r(t) by (9.11)In short, we have proved the following result.

Moreover the solution of these variational equations is
Now we compute the solution of the variational equations (9.7) of the Kepler problem (9.2) along the elliptic solution (r(t), R(t)) with initial conditions r(0) = (1 ± e)/2 and R(0) = 0.
We analyze here the solution of the variational equations of the elliptic Sitnikov problem (9.3) along the solution curve (z(t), Z(t)) for e > 0 sufficiently small.These variational equations are given by the matrix differential equation with initial condition A 4 (0) = I (the 2 × 2 identity matrix), where and Thus the derivatives (∂z/∂z 0 , ∂Z/∂z 0 ) and (∂z/∂Z 0 , ∂Z/∂Z 0 ) are given by the solution of system with initial conditions x(0) = 1, y(0) = 0 and x(0) = 0, y(0) = 1, respectively.By the Poincaré expansion theorem this solution may be expanded in power series of e and ) is the solution of the variational equations (9.9) of the circular Sitnikov problem along the solution curve (z (0) (t), Z (0) (t)) and We remark that the solution of the variational equations (9.9) of the circular Sitnikov problem along the solution curve (z (0) (t), Z (0) (t)) is unbounded when t goes to infinity.Therefore, with a fixed value of e, (9.23) is valid only for t less than a constant which depends on the value of e.

Continuation of periodic orbits from the reduced circular Sitnikov problem to the reduced isosceles problem.
In this section we will use the analytic continuation method of Poincaré to continue the periodic orbits of the reduced circular Sitnikov problem to symmetric periodic orbits of the reduced isosceles problem for µ > 0 sufficiently small.Choosing conveniently the origin of time, the periodic orbit of the reduced circular Sitnikov problem with period T > π/ √ 2 is the orbit associated to the periodic solution with initial conditions ϕ 1/4 (t; r 0 = 1/2, ṙ0 = 0, z 0 = 0, ż0 = ż * 0 = √ 2h + 4, µ = 0).Here h ∈ (−2, 0) is the energy of the periodic orbit of period T ∈ (π/ √ 2, ∞) (see Theorem 7.5 for details).We remark that the notation used here is the one defined in section 8.
Since the reduced isosceles problem is autonomous, if we continue using different initial conditions defining the same periodic orbit, then we will obtain the same continued periodic orbits.So, it will be sufficient to continue periodic solutions with initial conditions ϕ 1/4 (t; 1/2, 0, 0, ż * 0 , 0) for −2 < h < 0. We note that these periodic solutions are doubly symmetric, so we can investigate their continuation to periodic solutions of the reduced isosceles problem for µ > 0 small that are either doubly symmetric, r-symmetric, or t-symmetric.Here we analyze only the continuation to doubly symmetric periodic solutions.We have also analyzed the continuation to rand to t-symmetric periodic solutions, but these two types of continuation provide again the same families of doubly symmetric periodic orbits of the reduced isosceles problem for µ > 0 small (for details see [6]).
Observe that τ = T = T (h), r 0 = 1/2, ż0 = ż * 0 = √ 2h + 4, and µ = 0 is a solution of (10.1) for each −2 < h < 0. It corresponds to the doubly symmetric periodic solution ϕ 1/4 (t; 1/2, 0, 0, ż * 0 , 0) of the reduced circular Sitnikov problem.Our aim is to find solutions of (10.1) near the known solution τ = T , r 0 = 1/2, ż0 = ż * 0 , and µ = 0.For this goal, we will apply the implicit function theorem to (10.1) in a neighborhood of that point, choosing (r 0 , ż0 ) as the dependent variables and (µ, τ ) as the independent ones.Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We note that there are five other choices for the dependent (independent) ables.Since we want to continue periodic solutions from µ = 0 to µ > 0 small, we are interested in solutions of (10.1) depending on µ.So, the other possible choices for the independent variables are (µ, r 0 ) and (µ, ż0 ).Since the reduced isosceles problem possesses the first integral of the energy, we also could be interested in expressing the solutions of (10.1) as a function of µ and of the energy h.We have analyzed these other possible choices for the independent variables, then saw that the implicit function theorem using either (µ, r 0 ) or (µ, h) as the independent variables cannot be applied to this problem because the corresponding determinant vanishes.Moreover, if we apply the implicit function theorem using (µ, ż0 ) as the independent variables, we obtain the same solutions of (10.1) as we do using (µ, τ ).The difference is that these solutions are parameterized by (µ, ż0 ) instead of (µ, τ ).
In short, if the period T = T (h) is a nonmultiple of 4π, then determinant (10.2) is different from zero.This proves the following theorem.Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpTheorem 10.1.For any T > π/ √ 2, with T = 4πn for all n ∈ N, the periodic orbit of the reduced circular Sitnikov problem with period T can be continued to a 2-parameter family (on µ and τ ) of doubly symmetric periodic orbits of the reduced isosceles problem (3.1) with angular momentum c = 1/4, which have period τ for (µ, τ ) in a sufficiently small neighborhood of (0, T ) with µ 0. 10.1.Remarks.We note that Theorem 10.1 also gives periodic orbits of the reduced isosceles problem for µ = 0.One might think that this theorem could be used to find new symmetric periodic orbits of the reduced elliptic Sitnikov problem.But this is not the case because the symmetric periodic orbits for µ = 0 that we obtain in this way are periodic orbits of the reduced circular Sitnikov problem, which are already known.This follows from the fact that the functions r 0 (µ, τ ) and ż0 (µ, τ ) are unique and that ϕ 1/4 (t; r 0 = 1/2, 0, 0, ż0 = 2h(τ ) + 4, 0) is a periodic solution of the reduced circular Sitnikov problem.
On the other hand, Theorem 10.1 does not allow us to continue the periodic orbits of the reduced circular Sitnikov problem that have period T that is a multiple of 4π.Later on, in section 12, we will see that these periodic solutions can be continued in two steps to two different families of doubly symmetric periodic solutions of the reduced isosceles problem (3.1) with angular momentum c = 1/4 and µ > 0 sufficiently small, having period τ near T (see Theorem 12.8).The fact that the continuation is to two families explains why we have not been able here to continue these periodic orbits using only the implicit function theorem.
Often when we analyze a problem of continuation of periodic solutions we are interested in families of periodic solutions with the same period or with the same energy (these last families are called isoenergetic families).We could also consider families of periodic solutions with a fixed initial condition.In order to obtain these kinds of families in our problem we would fix one of the variables (it could be T , h, r 0 , or ż0 ) in system (10.1), and then we would continue, in function of µ, the known periodic solutions of the reduced circular Sitnikov problem.We have done that and seen that the periodic solutions of the reduced circular Sitnikov problem with period T , nonmultiple of 4π, can be continued to a 1-parameter family (on µ) of doubly symmetric periodic solutions of the reduced isosceles problem having fixed period T , and another 1-parameter family having fixed initial condition ż0 = ż * 0 .Clearly these two families are contained in the 2-parameter family of doubly symmetric periodic orbits of the reduced isosceles problem obtained in Theorem 10.1.Finally, the continuation fixing either the initial condition r 0 or the energy h is not possible because the corresponding determinants vanish.
Theorem 10.1 is improved by the following result.Theorem 10.2.
Proof.Fixed τ * ∈ [T 1 , T 2 ], the implicit function theorem assures the existence of two unique analytic functions r 0 (µ, τ ) and ż0 (µ, τ ) for (µ, τ ) in a sufficiently small neighborhood of (0, τ * ).Due to the compactness of [T 1 , T 2 ] and the uniqueness of r 0 (µ, τ ) and ż0 (µ, τ ), we can find µ 0 > 0 such that, for 0 µ < µ 0 , these functions are defined for all τ * ∈ [T 1 , T 2 ], which proves the result.Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php11.Continuation of symmetric periodic orbits from the reduced elliptic Sitnikov problem to the reduced isosceles problem.In this section we will continue the known symmetric periodic solutions of the reduced elliptic Sitnikov problem with eccentricity e (meaning the symmetric periodic solutions given in section 8) to symmetric periodic solutions of the reduced isosceles problem with c = c e and µ > 0 sufficiently small.In particular we will prove the following result.
Theorem 11.1.Let γ e0 be a symmetric periodic orbit of the reduced elliptic Sitnikov problem with eccentricity e 0 given by Theorems 8.2 or 8.3 that has period τ = 2πp = qT for fixed values of p, q ∈ N coprime with p > q/(2 √ 2).If the eccentricity e 0 is sufficiently small, then γ e0 can be continued to a 2-parameter family (on µ and τ ) of symmetric periodic orbits of the reduced isosceles problem (3.1) with angular momentum c = √ 1 − e 0 2 /4 and µ 0 that have period τ for (µ, τ ) in a sufficiently small neighborhood of (0, τ ).Moreover the continued periodic orbits satisfy the same symmetry as the initial orbit γ e0 .
Apart from the symmetric periodic orbits of the reduced elliptic Sitnikov problem given by Theorems 8.2 and 8.3, we know the existence of infinitely many symmetric periodic orbits of the reduced elliptic Sitnikov problem for all 0 < e < 1 (see Propositions 12 and 15 in [7]); unfortunately we do not know analytical expressions for their initial conditions.Nevertheless we will give sufficient conditions in order to continue an arbitrary symmetric periodic orbit of the reduced elliptic Sitnikov problem to symmetric periodic orbits of the reduced isosceles problem for µ > 0 sufficiently small.
We start analyzing the continuation of doubly symmetric periodic orbits of the reduced elliptic Sitnikov problem, after which we will analyze the continuation of rand t-symmetric periodic orbits.
Choosing conveniently the origin of time, the symmetric periodic orbits of the reduced elliptic Sitnikov problem can be seen as the orbits associated to symmetric periodic solutions with initial conditions either ϕ ce (t; r 0 = r 0 = (1 ± e)/2, ṙ0 = 0, z 0 = 0, ż0 = ż 0 , µ = 0) or ϕ ce (t; r 0 = r 0 = (1 ± e)/2, ṙ0 = 0, z 0 = z 0 , ż0 = 0, µ = 0).So, we will study only the continuation of symmetric periodic solutions of these types.Of course, if we continue different initial conditions defining the same periodic orbit, then we will obtain the same periodic orbit of the reduced isosceles problem.
The derivatives that appear in this determinant are obtained by evaluating at time t = τ /4 the corresponding solution of the variational equations of the reduced restricted isosceles problem (6.2) for c = c e along the solution curve ϕ ce (t; r 0 , 0, 0, ż 0 , 0) with r 0 = (1 ± e)/2.The solution of these variational equations has been studied in section 9.1.
We note that in order to continue the symmetric periodic solutions of the reduced elliptic Sitnikov problem, we applied the implicit function theorem, choosing µ and τ as the independent variables.As happened in the continuation of periodic solutions from the reduced circular Sitnikov problem (see section 10), there are other possible choices for the independent variables.These other possible choices are (µ, r 0 ), (µ, ż0 ), and (µ, h) (respectively, (µ, r 0 ), (µ, z 0 ), (µ, h)) when the starting initial condition that we continue is r-symmetric (respectively, t-symmetric).Here h is the energy of the periodic solution.We have analyzed these choices for the independent variables, but we have not obtained new periodic orbits.In particular, we have seen that the determinant that we must evaluate when we use (µ, ż0 ) (respectively, (µ, z 0 )) as the independent variables is more complicated than in the other cases because we do not know an explicit expression of some of the derivatives.
In particular, we also have analyzed the continuation of the symmetric periodic solutions of the reduced elliptic Sitnikov problem given by Theorems 8.2 and 8.3 to symmetric periodic solutions of the reduced isosceles problem by fixing either the period, one of the initial conditions, or the energy.We have seen that if the eccentricity e is sufficiently small, then these symmetric periodic solutions can be continued to families of symmetric periodic solutions of the reduced isosceles problem for µ > 0 sufficiently small that have either the same period, the same initial condition r 0 , or the same energy h as the initial orbit.We have also evaluated numerically for some periodic orbits the correspondent determinant when we continue by fixing the initial condition ż0 (respectively, z 0 ), and we have seen that it is different from zero.Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpcontinuation in two steps also give different periodic orbits.Finally the periodic orbits of the reduced circular Sitnikov problem with period T = 2πω, where ω > 1/(2 √ 2) is an irrational number, can be continued by direct continuation, but they cannot be continued in two steps.
13. Summary.The main results about continuation of the periodic orbits of the reduced circular Sitnikov problem to symmetric periodic orbits of the reduced isosceles problem for µ > 0 sufficiently small-that is, Theorem 10.1 and Theorem 12.8-are summarized in Theorem A of the introduction.
In Remark 12.2 we have seen that we can work with the parameter τ = 2πp f (e) (the period) instead of the eccentricity e.Thus the 2-parameter families of periodic orbits of the reduced isosceles problem obtained from continuation in two steps of periodic orbits of the reduced circular Sitnikov problem with period T = 2πp/q for p, q ∈ N coprime with p > q/(2 √ 2) can be parameterized by means of µ and τ instead of µ and e.This means that Theorem A of the introduction can be stated using µ and τ as parameters instead of µ and e.
Next we give the extension of Theorem A to the full isosceles problem (see section 4 for more details about the relationship between the periodic orbits of the reduced isosceles problem and the orbits of the full isosceles problem).
Let Π T denote the two-dimensional invariant torus of the restricted isosceles problem that comes from a periodic orbit of the reduced circular Sitnikov problem with period T .Then we have the following result.
Theorem 13.1.The torus of the circular restricted isosceles problem Π T with T > π/ √ 2 can be continued to the following families of two-dimensional tori of the isosceles problem with µ > 0 sufficiently small.These tori are filled with either periodic or quasi-periodic orbits: 1. Case T = 2πω with ω > 1/(2 √ 2) an irrational number.(a) Π T can be continued directly to one 2-parameter family (on µ and τ with τ sufficiently close to T ) of two-dimensional tori.2. Case T = 2πp/q for some p, q ∈ N coprime with p > q/(2 √ 2).(a) p odd: i. Π T can be continued directly to one 2-parameter family (on µ and τ with τ sufficiently close to T ) of two-dimensional tori.ii.Π T can be continued by two steps to four 2-parameter families (on µ and τ with τ sufficiently close to T q) of two-dimensional tori.(b) p even and q = 1: i. Π T can be continued directly to one 2-parameter family (on µ and τ with τ sufficiently close to T ) of two-dimensional tori.ii.Π T can be continued by two steps to two 2-parameter families (on µ and τ with τ sufficiently close to T q) of two-dimensional tori.(c) p even and q = 1: i. Π T can be continued by two steps to two 2-parameter families (on µ and τ with τ sufficiently close to T q) of two-dimensional tori.By Proposition 7.7, the tori Π T are filled with periodic orbits when T = p2π/q for some p, q ∈ N coprime with p > q/(2 √ 2); and they are filled with quasi-periodic orbits when T = 2πω with ω > 1/(2 √ 2) an irrational number.So, in particular, we have continued tori filled with quasi-periodic orbits.The tori of the isosceles problem for µ > 0 that we have obtained are filled with either periodic or quasi-periodic orbits of the isosceles problem.
Remember that the phase portrait of the isosceles problem on each angular mo-Downloaded 09/25/12 to 150.128.148.9.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpmentum level c with c = 0 is the same (see Proposition 3.1).Therefore we have obtained invariant periodic and quasi-periodic two-dimensional tori on each angular momentum level c = 0.

11.3.1. Application of Theorem 11.6. Now
we apply Theorem 11.6 to continue the t-symmetric periodic solutions of the reduced elliptic Sitnikov problem given by Theorem 8.3.Let ϕ ce (t; r 0