Periodic orbits of the planar collision restricted 3-body problem

PERIODIC ORBITS OF THE PLANAR COLLISION RESTRICTED 3–BODY PROBLEM JAUME LLIBRE1AND DANIEL PAS¸CA2,3 Abstract. Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3–body problem. Additionally, we also continue to this restricted problem the so called “comets orbits”. 1. Introduction We consider a special case of the restricted 3–body problem, called collision restricted 3–body problem. In our problem the two primaries with equal masses are moving in a collision orbit of the 2–body problem, and the infinitesimal mass is moving on a plane orthogonal to the line of motion of the primaries. The purpose of this paper is to find symmetric periodic orbits of this problem using the continuation method developed by Poincar´e which is one of the most frequently used methods for proving the existence of periodic orbits. Starting with Poincar´e in [8], which used this method to prove the existence of periodic orbits for circular planar restricted 3–body problem, see [10], this method was used also by other authors in different problems like spatial isosceles 3–body problem (see [1], [5]), circular and elliptic Sitnikov problem (see [3], [4]), collinear restricted 3–body problem (see [2]), etc. In [6], we find a good discussion on the applicability of the Poincar´e’s continuation method to different N–body problems. We give the equations of motion of the collision restricted 3–body problem in Section 2. In Section 4 we introduce three different kinds of symmetric periodic orbits. In section 3, 5 and 6 we will prove the continuation of the circular orbits (see Theorem 1), of the different three kinds of symmetric 1991 Mathematics Subject Classification. Primary 70F15(70F07). Key words and phrases. Collision restricted 3–body problem, continuation method, periodic orbits. †The first author is partially supported by a DGICYT grant number BFM 2002- 04236-C02-02 and by a CICYT grant number 2001SGR00173. The second author was supported by Ministerio de Educacio´n y Ciencia, grant number SB2003-0337. 1 2 JAUME LLIBRE AND DANIEL PAS¸CA elliptic orbits (see Theorems 5, 6, 7) and of the comets orbits (see Theorem 8) from the rotating Kepler problem to the collision restricted 3–body problem. 2. The collision restricted 3–body problem Let m1=m2be two punctual masses (called primaries) describing a collision orbit of the 2–body problem. We consider an infinitesimal mass m3that moves on the plane Π orthogonal to the line of motion of the primaries through their center of mass. We choose the units of mass, length and time so that m1=m2= 1/2, the gravitational constant G= 1, and the period of the two primaries is 2π. We take the origin of coordinates at the center of mass, the plane of the motion of the infinitesimal body is the (X1, X2)–plane and the motion of the primaries take place on the z–axis (see Figure 1). Then the equations of motion of the infinitesimal mass are ¨ X1=−X1 (X2 1+X2 2+z2(t))3/2, ¨ X2=−X2 (X2 1+X2 2+z2(t))3/2, where z(t) is the solution of the elliptic collision 2–body problem, which we take as z(t) = µ(1 −cos E(t)), where µ > 0 is a small parameter and E(t) is the eccentric anomaly which is a function of time via Kepler’s equation E−sin E=t. We note that since Eis 2π–periodic, zwill be 2π–periodic. For more details on the motion of the primaries see [9]. We want to analyze the continuation of periodic solutions for this problem from µ= 0 to µ > 0 small. Like in the case of planar circular restricted 3–body problem, we cannot prove the continuation of the periodic solutions in fixed coordinates because the system in these coordinates is very degenerated. For this reason, we consider a rotating coordinate system (x1, x2, z) related with the fixed one through the following formula   x1 x2 z  =  cos tsin t0 −sin tcos t0 0 0 1    X1 X2 z  . PERIODIC ORBITS OF A COLLISION RESTRICTED 3–BODY PROBLEM 3 m1= 1/2 (0,0, z(t)) m2= 1/2 (0,0,−z(t)) m3= 0 (X1, X2,0) X1 X2 z O Π Figure 1. The elliptic collision 2–dimensional restricted 3–body problem The Hamiltonian governing the motion of the infinitesimal mass for this collision restricted 3–body problem is (1) H=kyk2 2−xTKy −1 [kxk2+z2(t)]1/2, where x= (x1, x2) and y= (y1, y2) are conjugate, K=0 1 −1 0and xTKy is just the angular momentum. Since µis a small parameter the Hamiltonian can be written into the form (2) H=kyk2 2−xTKy −1 kxk+O(µ2). When µ= 0, His the Hamiltonian of the 2–body problem, or Kepler problem, in rotating coordinates. There are two main differences between the Hamiltonian given by (2) and the corresponding Hamiltonian in the 4 JAUME LLIBRE AND DANIEL PAS¸CA planar circular restricted 3–body problem. The first one is that for the collision restricted 3–body problem the perturbation is of the order µ2instead of the order µfor the planar circular restricted 3–body problem. The second is that our Hamiltonian (2) depends explicitly on the time (i.e. it is non-autonomous) but is 2π–periodic. The dependence is contained in the terms of O(µ2). 3. The continuation of the circular orbits First we change the coordinates from the cartesian coordinates (x1, x2, y1, y2) to the polar coordinates (r, θ, R, Θ). This change of coordinates is given by x1=rcos θ, x2=rsin θ y1=Rcos θ−Θ rsin θ, y2=Rsin θ+Θ rcos θ. For more details see [7]. In these new coordinates the Hamiltonian is of the form (3) H=1 2R2+Θ2 r2−Θ−1 r+O(µ2). Now, following Section C of Chapter VI of [7], we can prove in the same way the continuation of the circular orbits (or periodic orbits of the first kind in the terminology introduced by Poincar´e; for more details see [10]) from µ= 0 to µ > 0 sufficiently small. Theorem 1. Let p,qrelatively prime integers and T= 2πp/q. Then the circular T–periodic orbits of the Kepler problem in rotating coordinates with angular momentum ccan be continued to a periodic orbit of period 2πp of the collision restricted 3–body problem for small values of µ > 0. Proof. For µ= 0 the equations of motion are ˙r=R, ˙ R=Θ2 r3−1 r2, ˙ θ=Θ r2−1,˙ Θ=0. Let Θ = cbe a fixed constant. The circular orbit R= 0, r=c2is a periodic solution with period |2πc3/(1 −c3)|(this is the time for θto increase by 2π). We wish to consider the case when |2πc3/(1 −c3)|=2πp q, PERIODIC ORBITS OF A COLLISION RESTRICTED 3–BODY PROBLEM 5 which implies that c=p p+q1/3or c=p p−q1/3. Linearizing the rand Requations about this solution gives ˙r=R, ˙ R=−c−6r, which has solutions of the form exp(±it/c3), and so the nontrivial multipliers of the circular orbits are exp(±i2π/(1 −c3)) which are not +1, since 1/(1 −c3) is not an integer. We apply the results of the Appendix to the Hamiltonian system defined by (3) thinking this system as a 2πp–periodic system, and continuing the circular orbit of period 2πp/q rotated qtimes. Then, this orbit can be continued for µ > 0 small. 4. Symmetric periodic orbits The equations of motion of the infinitesimal mass in the rotating coordinates are ¨x1=x1+ 2 ˙x2−x1 (x2 1+x2 2+z2(t))3/2, ¨x2=x2−2 ˙x1−x2 (x2 1+x2 2+z2(t))3/2. (4) It is easy to check that the equations of motion (4) are invariant under the symmetry (t, x1,˙x1, x2,˙x2)−→ (−t, x1,−˙x1,−x2,˙x2). This means that if ϕ(t) = (x1(t),˙x1(t), x2(t),˙x2(t)) is a solution of system (4), then also ψ(t) = (x1(−t),−˙x1(−t),−x2(−t),˙x2(−t)) is a solution. We note that this symmetry corresponds to a symmetry with respect to the x1–axis, and will be denoted by S1. In fact, is well known the following result. Proposition 2. Let ϕ(t) = (x1(t),˙x1(t), x2(t),˙x2(t)) be a solution of system (4). If ˙x1(t)and x2(t)are zero at t=t0and at t=t0+T/2but are not simultaneously zero at any value of t∈(t0, t0+T/2), then ϕ(t)is a S1– symmetric periodic solution of period T. Equations (4) are also invariant under the symmetry (t, x1,˙x1, x2,˙x2)−→ (−t, −x1,˙x1, x2,−˙x2), i.e., the symmetry with respect to the x2–axis, which will be denoted by S2 and is characterized as follows. 6 JAUME LLIBRE AND DANIEL PAS¸CA Proposition 3. Let ϕ(t) = (x1(t),˙x1(t), x2(t),˙x2(t)) be a solution of system (4). If x1(t)and ˙x2(t)are zero at t=t0and at t=t0+T/2but are not simultaneously zero at any value of t∈(t0, t0+T/2), then ϕ(t)is a S2– symmetric periodic solution of period T. We note that there could be periodic solutions of (4) that are simultaneously S1– and S2–symmetric. These periodic solutions will be called doubly symmetric periodic solutions and are characterized by the following result. Proposition 4. Let ϕ(t) = (x1(t),˙x1(t), x2(t),˙x2(t)) be a solution of system (4). (a) If ˙x1(t)and x2(t)are zero at t=t0and x1(t)and ˙x2(t)are zero at t=t0+T/4but are not simultaneously zero at any value of t∈(t0, t0+T/4), then ϕ(t)is a doubly symmetric periodic solution of period T. (b) If x1(t)and ˙x2(t)are zero at t=t0and ˙x1(t)and x2(t)are zero at t=t0+T/4but are not simultaneously zero at any value of t∈(t0, t0+T/4), then ϕ(t)is a doubly symmetric periodic solution of period T. For the proof of the above propositions, see for instance [3]. 5. The continuation of some symmetric elliptic orbits Poincar´e defines periodic orbits of the second kind as those which are generated from elliptic 2–body orbits. Again for more details see [10]. To prove the continuation of the circular and symmetric elliptic orbits we change the coordinates from the polar coordinates (r, θ, R, Θ) to Delaunay’s coordinates (l, g, L, G). We recall here the geometrical meaning of these coordinates -lis the mean anomaly and is measured from the perihelion; -gis the argument of the pericenter and determines the orientation of the major semi-axis of the ellipse in the plane of motion; -Lis related to the major semi-axis, a, by L=a1/2; -Gis related to the major semi-axis, a, and to the eccentricity, e, by G= [a(1 −e2)]1/2. In Delaunay coordinates the Hamiltonian (3) takes the form (5) H=−1 2L2−G+O(µ2). Following [11], we can prove in a similar way to the circular planar restricted 3–body problem, the continuation to µ > 0 small of the elliptic orbits for the collision restricted 3–body problem. Taking into account the PERIODIC ORBITS OF A COLLISION RESTRICTED 3–BODY PROBLEM 7 possible symmetries of the periodic solutions we have three different situations. 5.1. Continuation of S1–symmetric periodic orbits. We start to remark that we have 8 different S1–symmetric elliptic orbits for the Kepler problem in rotating coordinates which can be continued into S1–symmetric periodic orbits of the collision restricted 3–body problem when µ > 0 is small. These different orbits are easy to describe in terms of the initial conditions in Delaunay coordinates g, l. For the counter-clockwise motion of the infinitesimal mass we can have the following initial conditions (a) g(0) = 0, l(0) = 0, which means that the pericenter is on the positive x1–semiaxis and the infinitesimal mass is at the pericenter; (b) g(0) = π,l(0) = 0 which means that the pericenter is on the negative x1–semiaxis and the infinitesimal mass is at the pericenter; (c) g(0) = 0, l(0) = πwhich means that the pericenter is on the positive x1–semiaxis and the infinitesimal mass is at the apocenter; (d) g(0) = π,l(0) = πwhich means that the pericenter is on the negative x1–semiaxis and the infinitesimal mass is at the apocenter. There are other 4 different S1–symmetric elliptic orbits for the clockwise motion of the infinitesimal mass. Now we analyze the situation when the motion of the infinitesimal body is counter-clockwise. In Delaunay coordinates (l, g, L, G), an orthogonal crossing of the x1–axis at a time t0is described by (6) l(t0) = n1π, g(t0) = n2π, n1, n2integers. In order to have a S1–symmetric orbit we need to have 2 different orthogonal crossings of the x1–axis. We will prove that, mod 2πin time, the 4 S1– symmetric elliptic orbits can be continued into 16 different S1–symmetric orbits of the collision restricted 3–body problem when µis small. These 16 possibilities come from the fact that we can have the first orthogonal crossings of the x1–axis in any of the situation (a), (b), (c) or (d) and the second orthogonal crossings of the x1–axis also in any of this 4 different situations. In the situation when the motion of the infinitesimal body is clockwise we will obtain other 16 different S1–symmetric periodic orbits of the collision restricted 3–body problem. Theorem 5. Let p, q be relatively prime integers and T= 2πp/q. Then the elliptic T–periodic orbit of the Kepler problem in rotating coordinates which satisfies (7) l(0) = n1π, g(0) = n2π, L3(0) = p/q, G(0) = constant, 8 JAUME LLIBRE AND DANIEL PAS¸CA can be continued into a S1–symmetric periodic of period 2πp orbit of the collision restricted 3–body problem for small values of µ > 0. Proof. The Hamiltonian of the collision restricted 3–body problem in Delaunay coordinates for small µis given by (5) and the equations of motion are ˙ l=1 L3+O(µ2),˙ L= 0 + O(µ2), ˙g=−1 + O(µ2),˙ G= 0 + O(µ2). (8) For µ= 0 we consider the elliptic orbit satisfying the initial conditions (7). The motion on this ellipse is l=tq/p +n1π, L = (p/q)1/3, g=−t+n2π, G = constant. The period of the infinitesimal mass following this ellipse is T= 2πL3= 2πp/q. Thus, the period of the motion of the three body together is T′= 2πp, so g(T′/2) = (−p+n2)πand l(T′/2) = (q+n1)π. Hence, that orbit satisfies (6) (and is thus S1–symmetric). Now for small µ > 0 we consider solutions satisfying (8) and the initial conditions l(0) = n1π, g(0) = n2π, L(0) = L0, G(0) = G0. We wish to solve ψ1(L0, G0, t, µ) = 0, ψ2(L0, G0, t, µ) = 0, for L0near (p/q)1/3and tnear T′/2 where ψ1(L0, G0, t, µ) = g(t) + (p−n2)π, and ψ2(L0, G0, t, µ) = l(t)−(q+n1)π. By the Implicit Function Theorem, it suffices to show that det ∂ψ1/∂t ∂ψ2/∂t ∂ψ1/∂L0∂ψ2/∂L06= 0 at the elliptic orbit, but for µ= 0 we have det ∂g/∂t ∂l/∂t ∂g/∂L0∂l/∂L0= det −1q/p 0−3t(q/p)4/36= 0, for any t > 0. PERIODIC ORBITS OF A COLLISION RESTRICTED 3–BODY PROBLEM 9 We want to remark that some of these different families of S1–symmetric periodic orbits are doubly symmetric but we will consider these situations later on. 5.2. Continuation of S2–symmetric periodic orbits. Like in previous subsection we have 8 different S2–symmetric elliptic orbits for the Kepler problem in rotating coordinates which can be continued into the S2– symmetric orbits of the collision restricted 3–body problem when µis small. Again, these different orbits are easy to describe in terms of the initial conditions in Delaunay coordinates g, l. For the counter-clockwise motion of the infinitesimal mass we can have the following initial conditions (a) g(0) = −π/2, l(0) = 0, which means that the pericenter is on the negative x2–semiaxis and the infinitesimal mass is at the pericenter; (b) g(0) = −π/2, l(0) = πwhich means that the pericenter is on the negative x2–semiaxis and the infinitesimal mass is at the apocenter; (c) g(0) = π/2, l(0) = πwhich means that the pericenter is on the positive x2–semiaxis and the infinitesimal mass is at the apocenter; (d) g(0) = π/2, l(0) = 0 which means that the pericenter is on the positive x2–semiaxis and the infinitesimal mass is at the pericenter. There are other 4 different S1–symmetric elliptic orbits for the clockwise motion of the infinitesimal mass. We start to analyze the situation when the motion of the infinitesimal body is counter-clockwise. In Delaunay coordinates (l, g, L, G), an orthogonal crossing of the x2–axis at a time t0is described by (9) l(t0) = n1π, g(t0) = n2π+π/2, n1, n2integers. and to have an S2–symmetric orbit we need to have 2 different orthogonal crossings of the x2–axis. We will see that, mod 2πin time, the 4 elliptic orbits can be continued into 16 different S2–symmetric periodic orbits of the collision restricted 3–body problem when µis small. These 16 possibilities come from the fact that we can have the first orthogonal crossings of the x2–axis in any of the situation (a), (b), (c) or (d) and the second orthogonal crossings of the x2–axis also in any of this 4 different situations. In the situation when the motion of the infinitesimal body is clockwise we will obtain other 16 different S2–symmetric orbits of the collision restricted 3–body problem. Theorem 6. Let p, q be integers and T= 2πp/q. Then the elliptic T– periodic orbit of the Kepler problem in rotating coordinates which satisfies (10) l(0) = n1π, g(0) = n2π+π/2, L3(0) = p/q, G(0) = constant,