Infinitely many periodic orbits for the rhomboidal five-body problem

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1 Infinitely many periodic orbits for the rhomboidal five-body problem Montserrat Corbera and Jaume Llibre Citation: J. Math. Phys (006); doi:.63/ View online: View Table of Contents: Published by the American Institute of Physics. Related Articles Dziobek configurations of the restricted (N + 1)-body problem with equal masses J. Math. Phys (01) Laser heating of finite two-dimensional dust clusters: B. Simulations Phys. Plasmas (01) Super central configurations of the three-body problem under the inverse integer power law J. Math. Phys (011) Thomas-forbidden particle capture J. Math. Phys (011) Scattering of scalar waves by many small particles AIP Advances (011) Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 JOURNAL OF MATHEMATICAL PHYSICS Infinitely many periodic orbits for the rhomboidal five-body problem Montserrat Corbera a Department of Digital and Information Technologies University of Vic Laura Vic Barcelona Spain Jaume Llibre b Department of Mathematics Autonomous University of Barcelona Bellaterra Barcelona Spain Received 0 July 006; accepted 5 October 006; published online 8 December 006 We prove the existence of infinitely many symmetric periodic orbits for a regularized rhomboidal five-body problem with four small masses placed at the vertices of a rhombus centered in the fifth mass. The main tool for proving the existence of such periodic orbits is the analytic continuation method of Poincaré together with the symmetries of the problem. 006 American Institute of Physics. DOI:.63/ I. INTRODUCTION In this paper we consider a particular case of the planar five-body problem defined as follows. We consider a mass m 0 =1 at the origin of coordinates with zero initial velocity two small masses m 1 =m = 1 with initial positions and velocities on the x axis symmetric with respect to the origin and two small masses m 3 =m 4 = with initial positions and velocities on the y axis also symmetric with respect to the origin see Fig. 1. Our five-body problem consists of describing the motion of the five masses under their mutual Newtonian gravitational attraction. Due to the symmetry of the initial conditions and velocities the four small bodies form a rhombus with center at m 0 at any time and the mass m 0 remains at rest at the origin. The description of the motion of this five-body problem is called the rhomboidal five-body problem. Although this is a five-body problem it can be formulated as a Hamiltonian system of two degrees of freedom one is the distance x0 ofm 1 to the origin and the other is the distance y 0 ofm 3 to the origin the distances of m and m 4 to the origin are obtained by symmetry. The system has three singularities the triple collision between m 0 m 1 and m the triple collision between m 0 m 3 and m 4 and the total collision of the five bodies. Due to the symmetries doing a double Levi-Civita transformation we regularize both triple collisions. When =0 the problem is reduced to two collision two-body problems the collision twobody problem with m 0 and m 1 and the collision two-body problem with m 0 and m 3. Note that if we take into account the five bodies then really for =0 we have instead of the binary collisions m 0 with m 1 and m 0 with m 3 the triple collisions m 0 m 1 and m and m 0 m 3 and m 4. Since the solutions of the collision two-body problem are known we can compute the periodic solutions of the regularized system for =0 in a fixed energy level h0. The objective of this paper is to prove that the symmetric periodic orbits of the regularized rhomboidal five-body problem for =0 can be continued to symmetric periodic orbits of the regularized rhomboidal five-body problem for 0 sufficiently small. The main tool for proving this result is the classical analytic continuation method of Poincaré. a Electronic mail: montserrat.corbera@uvic.cat b Electronic mail: jllibre@mat.uab.cat /006/471/1701/13/$ American Institute of Physics

3 1701- M. Corbera and J. Llibre J. Math. Phys FIG. 1. The rhomboidal five-body problem. The paper is structured as follows. In Sec. II we give the equations of motion of the rhomboidal five-body problem and we apply a double Levi-Civita transformation to regularize the triple collision between m 0 m 1 and m and the triple collision between m 0 m 3 and m 4. Notice that the total collision of the five bodies is not regularized. In Sec. III we analyze the discrete symmetries of the regularized problem. In particular we see that there are three different symmetries that provide symmetric periodic solutions of the problem. We characterize these symmetric periodic solutions and the double symmetric periodic solutions which are the main objective of this work. In Sec. IV we compute the periodic solutions of the regularized rhomboidal five-body problem for =0 and in particular we analyze the double symmetric periodic solutions. Finally in Sec. V we apply the analytic continuation method of Poincaré to continue the double symmetric periodic orbits of the regularized rhomboidal five-body problem for =0 to double-symmetric periodic orbits of the regularized rhomboidal five-body problem for 0 sufficiently small. II. EQUATIONS OF MOTION FOR THE RHOMBOIDAL FIVE-BODY PROBLEM We consider five point particles with masses m 0 =1 m 1 =m = 1 m 3 =m 4 = positions q 0 =00 q 1 =x0 q = x0 q 3 =0y and q 4 =0 y respectively and velocities v 0 =00 v 1 =v x 0 v = v x 0 v 3 =0v y and v 4 =0 v y respectively see Fig. 1. Our fivebody problem consist of describing the motion of these particles under their mutual Newtonian gravitational attraction. We note that due to the symmetry of the problem the mass m 0 is at rest at the origin and the motion of the masses m 1 and m respectively m 3 and m 4 is confined to the x axis respectively y axis. Since the configuration of the four bodies in motion is always a rhombus with center at m 0 we call the five-body problem the rhomboidal five-body problem. Without loss of generality we can assume that the gravitational constant is G=1. Then the kinetic energy of the rhomboidal five-body problem is T = 1 ẋ + ẏ where the dot denotes derivative with respect to the time t and the potential energy is U = x 4+ y 4 1 x + y. The Lagrangian of the problem is given by L=T U. By the Legendre transformation see for instance Refs. 1 3 the Hamiltonian of the problem is H = p x + p y x y x + y where p x and p y are the conjugate momenta. The equations of motion associated to the Hamiltonian H are ẋ = p x 1 ṗ x = x 4 1 x x + y 3/

4 Periodic orbits of the rhomboidal five-body J. Math. Phys ẏ = p y ṗ y = 4+ y 4 1 y x + y 3/. 1 Doing the rescaling of the variables x= X y= Y and t= 3 T and denoting the new variables XY T again by xyt system 1 becomes ẋ = p x 1 ṗ x = x 4 1 x x + y 3/ ẏ = p y ṗ y = 4+ y This system is also Hamiltonian with Hamiltonian 4 1 y x + y 3/. H = p x + p y x 4+ y 4 1 x + y We note that system has three singularities: x=0 that corresponds to triple collision between m 0 m 1 and m y=0 that corresponds to triple collision between m 0 m 3 and m 4 and finally x +y =0 that corresponds to the total collision of the five bodies. We regularize both triple collisions applying a double Levi-Civita transformation see Refs. 4 6 x = 1 y = p x = 1 1 p y = dt =4 1 ds. The regularized system of the rhomboidal five-body problem on the level energy H=h for some constant h is the Hamiltonian system d 1 ds = 1 1 d ds = 1 d 1 ds = 1 +8h / 6 3 with Hamiltonian d ds = 1 +8h / 6 K = h and satisfying the energy relation K=0; i.e. H=h. We note that system 3 is analytic with respect to its variables except when =0 which corresponds to the total collision. The regularization of the triple collisions allows us to look for periodic orbits of the rhomboidal five-body problem containing triple collisions between m 0 m 1 and m and between m 0 and m 3 and m 4. Our aim is to find periodic orbits of the rhomboidal five-body problem 3 for 0 sufficiently small satisfying the energy relation K=0. In fact we look only for symmetric periodic orbits which are easier to study than the general periodic orbits.

5 M. Corbera and J. Llibre J. Math. Phys III. SYMMETRIES It is easy to check that system 3 is invariant under the discrete symmetries Id: 1 1 s 1 1 s S 1 : 1 1 s 1 1 s S : 1 1 s 1 1 s S 3 : 1 1 s 1 1 s S 4 : 1 1 s 1 1 s S 5 : 1 1 s 1 1 s S 6 : 1 1 s 1 1 s S 7 : 1 1 s 1 1 s. The invariance under these symmetries means that if s= 1 s s 1 s s is a solution of system 3 then also S i s is a solution for i= An orbit s is called S i -symmetric if S i s=s. We note that IdS 1...S 7 with the usual composition forms an abelian group isomorphic to Z Z Z. This discrete group of symmetries appeared in many Hamiltonian systems as for instance the anisotropic Kepler problem 7 the Manev anisotropic problem 8 or the collinear threebody problem. 9 Using the uniqueness theorem of a solution of an ordinary differential system it follows easily that s is a S 1 -symmetric solution if and only if s intersects the plane 1 = =0 at least in one point. Now it is clear that a periodic solution is S 1 -symmetric if and only if it has exactly two intersection points with the plane 1 = =0. So clearly the half-period of such a S 1 -symmetric periodic orbit is the time which the orbit needs for travel from one of the intersection points to the other. Using similar arguments for the other symmetries we obtain the following proposition. Proposition 1: Let s= 1 s s 1 s s be a solution of system 3. a If 1 s and s are zero at s=s 0 and at s=s 0 +S/ but they are not simultaneously zero at any value of ss 0 s 0 +S/ then s isas 1 -symmetric periodic solution of period S. b If s and 1 s are zero at s=s 0 and at s=s 0 +S/ but they are not simultaneously zero at any value of ss 0 s 0 +S/ then s isas -symmetric periodic solution of period S. c If 1 s and s are zero at s=s 0 and at s=s 0 +S/ but they are not simultaneously zero at any value of ss 0 s 0 +S/ then s isas 3 -symmetric periodic solution of period S. Since in system 3 the total collision is not regularized in our study we must avoid the orbits of the rhomboidal five-body problem which start or end in total collision. In the variables that we are working the total collision takes place when 1 = =0. Therefore the symmetries S 4 and S 7 are not considered because their symmetric orbits present total collision. Due to the fact that S 5 =S 1 S 3 and S 6 =S S 3 studying the symmetric periodic orbits with respect to S 1 S and S 3 we shall get also the symmetric periodic orbits with respect to S 5 and S 6. There could be periodic solutions of system 3 that are simultaneously S 1 - and S -symmetric periodic solutions. These periodic solutions will be called S 1 -symmetric periodic solutions. Ina similar way we can define the S 13 -symmetric periodic solutions and the S 3 -symmetric periodic solutions. These kinds of symmetric periodic solutions are characterized in the following result.

6 Periodic orbits of the rhomboidal five-body J. Math. Phys Proposition : Let s= 1 s s 1 s s be a solution of rhomboidal five-body problem 3. a b c The solution s isas 1 -symmetric periodic solution of period S if and only if either 1 s 0 = s 0 =0 and s 0 +S/4= 1 s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that s= 1 s=0; or s 0 = 1 s 0 =0 and 1 s 0 +S/4= s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that 1 s= s=0. The solution s isas 13 -symmetric periodic solution of period S if and only if either 1 s 0 = s 0 =0 and 1 s 0 +S/4= s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that 1 s= s=0; or 1 s 0 = s 0 =0 and 1 s 0 +S/4= s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that 1 s= s=0. The solution s isas 3 -symmetric periodic solution of period S if and only if either s 0 = 1 s 0 =0 and 1 s 0 +S/4= s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that 1 s= s=0; or 1 s 0 = s 0 =0 and s 0 +S/4= 1 s 0 +S/4=0 and there is no ss 0 s 0 +S/4 such that s= 1 s=0. The next result shows that there are no symmetric periodic orbits with respect more than two symmetries. Proposition 3: There are no periodic solutions of the rhomboidal five-body problem 3 which are simultaneously S i -symmetric for i=13. Proof: Assume that s is a S i -symmetric periodic solution of period S for i=13. Then there exist times s 1 s and s 3 with s 1 s s 3 0S/ such that 1 s 1 = s 1 =0 s = 1 s =0 1 s 3 = s 3 =0. We assume that s 1 =0. This is not restrictive because system 3 is autonomous and consequently the origin of time can be chosen arbitrarily. Then since the orbit is in particular S 1 -symmetric from Proposition s =S/4. Similarly since it is also S 13 -symmetric again from Proposition s 3 =S/4. The fact that s =s 3 is a contradiction so we have proved the result. IV. SYMMETRIC PERIODIC SOLUTIONS FOR =0 For =0 system 3 becomes d 1 ds = 1 1 d 1 ds = 1 +8h d ds = 1 and the Hamiltonian K goes over to d ds = 1 +8h K = h 1. The Hamiltonian H for =0 can be written as H = H 1 xp x + H yp y = p x p y. x 4 y We note that H 1 x p x and H yp y are two fist integrals of the nonregularized problem so they are constant along the solutions in the intervals between two consecutive zeros of x and y. The flow of the rhomboidal five-body problem on the energy level H=h for some constant h is obtained from the flow of the Hamiltonian H 1 x p x on the energy level H 1 =h 1 and the flow of the Hamiltonian H yp y on the energy level H =h with h=h 1 +h. The Hamiltonian H 1 x p x in the Levi-Civita coordinates 1 1 is given by

7 M. Corbera and J. Llibre J. Math. Phys H 1 = = h 1 1 and the Hamiltonian H yp y in the Levi-Civita coordinates is H = 16 = h. Let 1 1 be a solution of system 4 satisfying the energy relation K=0 i.e. H=h we define a new time variable as follows: dt d =4 1. d ds = or equivalently The Hamiltonian K in the new time variable can be written as 5 K 1 = 1 K = h h 1 = h 1 1. Then 1 1 satisfy the system of differential equations associated to the Hamiltonian K 1 d 1 d = 1 1 d 1 d =8h 1 1. We are only interested in the periodic solutions of system 6. Thus we must consider only negative values of h 1. Then fixed h 1 0 system 6 can be integrated directly and the solution 1 1 of system 6 with initial conditions is 1 0 = * 1 0 = * 7 1 = cosw 1 + * sinw 1 w = cosw 1 w 1 1 sinw 1 8 where w 1 = h1 / 1. We note that the solution 8 is a periodic solution of system 6 with period =/w 1. Since we are interested in the periodic solution 8 satisfying the energy relation K 1 =0byEq.5 its period in the real time t is given by T 1 h 1 1 =0 Now we introduce a new time with 4 1 d =4 3/ 1 h 1. 6 d ds = 1 or equivalently dt d =4. Then are functions of the new time via the Hamiltonian system 9 with Hamiltonian d d = d d =8h

8 Periodic orbits of the rhomboidal five-body J. Math. Phys TABLE I. Period of s. Time t Time Time Time s T= pt 1 h 1 1 =qt h * = p * =q S * =st T/4 * /4 * /4 S * /4 K = 1 K = 8 4h 1 4. Moreover fixed h 0 the solution 1 of system with initial conditions is given by 0 = * 0 0 = * 0 11 = 0 cosw + * 0 sinw w = 0 cosw w 0 sinw 1 where w = h /. The solution 1 is periodic of period =/w. Moreover if the solution 1 satisfies the energy relation K =0 then by Eq. 9 the period of the solution 1 in the real time t is given by T h =0 4 d =4 3/ h. Proposition 4: Let 1 1 be a periodic solution of system 6 for a fixed h 1 0 with initial conditions 7 and period =/w 1 that satisfies K 1 =0. Let be the periodic solution of system for a fixed h 0 with initial conditions 11 and period =/w that satisfies K =0. Assume that h=h 1 +h and that s and s are given by Eqs. 5 and 9 respectively where we choose 0=0=0. Suppose that there is no sr such that 1 s = s=0. Then the following statements hold: a s= 1 s s 1 s s is a solution of system 4 with initial conditions 1 0= * 0= * 0 1 0= * and 0= * 0 that satisfies K=0. b If h 1 =p/q /3 1 h / for some pqn coprime then s is a periodic solution of system 4. c Assume that st is given by the inverse function of t= s d. Under the hypotheses of statement b the period and the quarter of the period of the periodic solution s using the different times t and s is given in Table I. Proof: Statement a follows easily from the definitions of 1 1 and together with the definitions of s and s. We have seen that in the time t 1 1 and are periodic solutions of periods T 1 h 1 1 and T h respectively. Thus in order to have a periodic solution of system 4 we need that pt 1 h 1 1 = qt h for some pqn coprime. Solving Eq. 13 with respect to h 1 we get that h 1 =p/q /3 1 h /. So statement b is proved. Now we see that the time t=t/4 corresponds to the time = * /4. In a similar way we can see that the time t=t/4 corresponds to the time = * /4 and s=s * /4. 13

9 M. Corbera and J. Llibre J. Math. Phys We note that system 6 is invariant under the symmetry This means that 1 = 1. So 1 is an even function. On the other hand it is easy to see that 1 is a periodic function of period /. Then from Eq. 5 we have that T 1 =0 4 1 d =0 / 4 1 d =40 /4 4 1 d. Moreover it is clear that Consequently 0 /4 p /4 / 4 1 d = /4 4 1 d = T 1 4. t * /4 4 1 =0 d = 4 1 p0 d = p T 1 4 = T 4. Therefore the time t=t/4 corresponds to = * /4. In short statement c is proved. We remark that the number p in Proposition 4 represents the number of triple collisions between m 0 m 1 and m during a period whereas q represents the number of triple collisions between m 0 m 3 and m 4. We are interested in symmetric periodic solutions of system 4 satisfying the energy relation K=0 with h=h 1 +h. In the next proposition we give initial conditions for those symmetric periodic solutions. Proposition 5: The following statements hold: a If p and q are odd then the solution s given by Proposition 4 with initial conditions /4 either * =0 * 0 = /h * =4 1 * 0 =0; b or * = 1 /h 1 * 0 =0 * =0 * 0 =4 ; isas 1 -symmetric periodic solution. If p is odd and q is even then the solution s given by Proposition 4 with initial conditions either * =0 * 0 = /h * =4 1 * 0 =0; c or * = 1 /h 1 * 0 = /h * =0 * 0 =0; isas 13 -symmetric periodic solution. If p is even and q is odd then the solution s given by Proposition 4 with initial conditions either * = 1 /h 1 * 0 =0 * =0 * 0 =4 ; isas 3 -symmetric periodic solution. or * = 1 /h 1 * 0 = /h * =0 * 0 =0; Proof: Solving K 1 =0 and K =0 for the initial conditions of the S 1 -symmetric periodic orbits given in Proposition a we get the initial conditions of statement a. In these initial conditions we have only considered the positive determination in the squareroots due to the fact that the Levi-Civita transformation duplicates the orbits. The proof follows from the evaluation of the solution s= 1 s s 1 s s with these initial conditions at times s=0 and

10 Periodic orbits of the rhomboidal five-body J. Math. Phys s=s * /4. We note that by Table I S * /4= 1 p /4 q /4 1 p /4 q /4. This completes the proof of statement a. The other statements follow similarly. V. CONTINUATION OF SYMMETRIC PERIODIC SOLUTIONS In this section using the continuation method of Poincaré see for instance Ref. we shall continue the symmetric periodic orbits of the rhomboidal five-body problem 3 from =0 to symmetric periodic orbits of system 3 for 0 sufficiently small. A. The S 1 -symmetric periodic solutions We denote by s;0 0 0= 1 s; 0 s; 0 1 s; 0 s; 0 the solution of 3 for fixed values of and h0 with initial conditions 1 0=0 0= 0 1 0= and 0=0. From Proposition a s;0 0 0 is a S 1 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relationk=0 if and only if S/4; 0 =0 1 S/4; 0 =0 K 0 =0. We solve equation K 0 =0 with respect the variable obtaining in this way = So s;0 0 0 is a S 1 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relation K=0 if and only if 14 S/4; 0 =0 1 S/4; 0 =0. 15 Notice that we have omitted the dependence with respect to which is given by Eq. 14. Assume that p= +1 q=k+1 for some kn0 and that h * 1 and h * verify that h =h * 1 +h * and h * 1 =p/q /3 1 h * /. By Propositions 4 and 5a we see that S=S * =spt 1 h * 1 =sqt h * 0 = * 0 = /h * is a solution of system 15 for =0. This solution correspond to the known S 1 -symmetric periodic solution s;0 * 0 * 00 of system 3 for =0 where * =4 1. Our aim is to continue this solution of system 15 for =0 to 0 sufficiently small. Applying the implicit function theorem to system 15 in a neighborhood of the known solution we have that if s 1 s s=s * /4 0 =* 0 =0 then we can find unique analytic functions 0 = 0 S=S defined for 0 sufficiently small such that i 0 0= * 0 S0=S * ii s;0 0 0 with 0 = 0 and given by 14 is a S 1 -symmetric periodic solution of system 3 with period S=S that satisfies the energy relation K=0. The derivatives /s and 1 /s evaluated at s=s * * /4 0 = 0 and =0 can be obtained directly from system 3 for =0 i.e. system 4 evaluating the right hand of the system at the solution s;0 * 0 * 00 at time s=s * /4. Then 0 16

11 1701- M. Corbera and J. Llibre J. Math. Phys and ss=s * /4 0 =* 0 =0 = 1 1 s=s * /4 1 * s=s * /4 s 0 =* 0 =0 0 = 0 =0 = 4 1k * h p q /3 0 = 8h 1 * 1 s=s * /4 0 =* 0 =0 It only remains to compute the value of 1 / 0 evaluated at s=s * /4 0 = * 0 =0. This value is given by the derivative evaluated at s=s * /4 and 0 = * 0 of the solution 1 s;0 0 * 00 with respect to 0 where 1 s;0 0 * 00 is the solution of system 4 with initial conditions 1 0=0 0= 0 1 0= * and 0=0 satisfying the energy relation K=0 see Proposition 4a. Then 1s;0 0 * 00s=S * /4 0 0s=S * /4. 0 =* 0 =0. = 1 s From Eqs. 5 and 9 we have that the times and are related by 1 d = d. 0 =* 0 Integrating this equation over the solutions 8 and 1 with the corresponding initial conditions and assuming that when 0=0=0 we have that s and s are related by the equation 4s 0 s sinw 1s 3 0 sinw s =0. 17 w 1 w 1 4w Since s;0 0 * 00 must be a solution of system 4 by Proposition 4 we have that K 1 =0 and K =0 so and w = h / with h = / 0 w 1 = h1 / 1 with h 1 = h h. Then derivating implicitly Eq. 17 with respect to 0 we obtain s s=s * /4 = q 0 0 =* q p /3 On the other hand from systems 6 and 8 we have that 1 s=s * /4 = 8h 1 1 s=s * /4 and Hence 0 =* 0 1 0s=S * /4 0 =* 0 0 =* = q p 1/3 h * = 1 ph * q p/3 * h.

12 Periodic orbits of the rhomboidal five-body J. Math. Phys s;0 0 * 00s=S * /4 0 0 =* 0 = h * p 1 + q p 1/3 q. So this derivative is different from zero. Hence the determinant 16 is not zero. In short we have proved the following result. Theorem 6: Given h0 and p and q odd positive integers the S 1 -symmetric periodic solution of the rhomboidal five-body problem 3 for =0 with initial conditions 1 0 =0 0= /h * 1 0=4 1 and 0=0 where h * =h /p/q /3 1 + can be continued to a -parameter family of S 1 -symmetric periodic orbits of the rhomboidal five-body problem 3for 0 sufficiently small. B. The S 3 -symmetric periodic solutions We denote by s; 0 00= 1 s; 0 s; 0 1 s; 0 s; 0 the solution of system 3 for fixed values of and h0 with initial conditions 1 0= 0= 0 1 0=0 and 0=0. From Proposition c s; 0 00 is a S 3 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relation K=0 if and only if S/4; 0 =0 1 S/4; 0 =0 K 0 =0. We solve equation K 0 =0 with respect to the variable 0 obtaining in this way 0 = 0. In particular 0 0= / 1 h. So s; 0 00 is a S 3 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relation K=0 if and only if S/4; =0 1 S/4; =0. 18 Assume that p= q=k+1 for some N and kn0 and that h * 1 and h * verify that h=h * 1 +h * and h * 1 =p/q /3 1 h * /. By Propositions 4 and 5c we see that S=S * =spt 1 h * 1 =sqt h * = * = 1 /h * 1 is a solution of system 18 for =0. This solution corresponds to the known S 3 -symmetric periodic solution s; * * of system 3 for =0 where 0 = * 0 = /h *. Our aim is to continue this solution of system 18 for =0 to 0 sufficiently small. Applying the implicit function theorem to system 18 in a neighborhood of the known solution we have that if s 1 s 1 s=s * /4 =* = then we can find unique analytic functions = S=S defined for 0 sufficiently small such that i 0= * S0=S * ii s; 0 00 with = and 0 = 0 is a S 3 -symmetric periodic solution of system 3 with period S=S that satisfies the energy relation K=0. Working as for the S 1 -symmetry we get that

13 M. Corbera and J. Llibre J. Math. Phys ss=s * /4 =* =0 0 and 1 s=s * /4 s =* =0 It only remains to compute the value of 1 / evaluated at s=s * /4 = * =0. This value is given by the derivative evaluated at s=s * /4 and = * of the solution 1 s; with respect to where 1 s; is the solution of system 4 with initial conditions 1 0= 0= =0 and 0=0 satisfying the energy relation K=0 see Proposition 4c. Then =0. 1s; s=s * /4 =* = 1 s + 1 s=s * /4 =* = h * p 1 p 1 q + q p 1/3 q so this derivative is different from zero. Hence the determinant 19 is not zero. In short we have proved the following result. Theorem 7: Given 1 0 0h0 p even and q odd positive integers the S 3 -symmetric periodic solution of the rhomboidal five-body problem 3 for =0 with initial conditions 1 0 = 1 /h 1 * 0= /h * 1 0=0 and 0=0 where h * =h /p/q /3 1 + h 1 * =hp/q /3 1 /p/q /3 1 + can be continued to a -parameter family of S 3 -symmetric periodic orbits of the rhomboidal five-body problem 3 for 0 sufficiently small. C. The S 13 -symmetric periodic solutions Let s; 0 00 be the solution of system 3 for fixed values of and h0 defined as in Sec. V B. From Proposition b s; 0 00 is a S 13 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relation K=0 if and only if 1 S/4; 0 =0 S/4; 0 =0 K 0 =0. Let 0 = 0 be the function defined in Sec. V B. Then s; 0 00 is a S 13 -symmetric periodic solution of the rhomboidal five-body problem with period S satisfying the energy relation K=0 if and only if 1 S/4; =0 S/4; =0. 0 Assume that p= +1 q=k for some N0 and kn and that h * 1 and h * verify that h=h * 1 +h * and h * 1 =p/q /3 1 h * /. By Propositions 4 and 5b we see that S=S * =spt 1 h * 1 =sqt h * = * = 1 /h * 1 is a solution of system 0 for =0. This solution correspond to the known S 13 -symmetric periodic solution s; * * of system 3 for =0 where 0 = * 0 = /h *. Our aim is to continue this solution of system 0 for =0 to 0 sufficiently small. Applying the implicit function theorem to system 0 in a neighborhood of the known solution we have that if

14 Periodic orbits of the rhomboidal five-body J. Math. Phys s s 1 s=s * /4 =* =0 then we can find unique analytic functions = S=S defined for 0 sufficiently small such that i 0= * S0=S * ii s; 0 00 with = and 0 = 0 is a S 13 -symmetric periodic solution of system 3 with period S=S that satisfies the energy relation K=0. Working as for the S 1 -symmetry we get that ss=s * /4 =* =0 0 and s=s * /4 s =* =0 It only remains to compute the value of / evaluated at s=s * /4 = * =0. This value is given by the derivative evaluated at s=s * /4 and = * of the solution s; with respect to where s; is the solution of system 4 with initial conditions 1 0= 0= =0 and 0=0 satisfying the energy relation K=0 see Proposition 4b. Then s; s=s * /4 =* = s + = 1k h p 1 + =0. s=s * /4 =* q p 1/3 q 1 +3 so this derivative is different from zero. Hence the determinant 1 is not zero. In short we have proved the following result. Theorem 8: Given h0 p odd and q even positive integers the S 13 -symmetric periodic solution of the rhomboidal five-body problem 3 for =0 with initial conditions 1 0 = 1 /h 1 * 0= /h * 1 0=0 and 0=0 where h * =h /p/q /3 1 + h 1 * =hp/q /3 1 /p/q /3 1 + can be continued to a -parameter family of S 13 -symmetric periodic orbits of the rhomboidal five-body problem 3 for 0 sufficiently small. ACKNOWLEDGMENTS The first author is partially supported by Grant No. MCYT-Spain MTM C0-01. The second author is partially supported by Grant Nos. MCYT-Spain MTM C0-01 and CIRIT-Spain 005SGR R. Abraham and J. E. Marsden Foundations of Mechanics Benjamin New York V. I. Arnold Mathematical Methods of Classical Mechanics Springer-Verlag Berlin K. R. Meyer and G. R. Hall An Introduction to Hamiltonian Dynamical Systems Springer-Verlag New York V. Szebehely Theory of Orbits Academic New York R. Broucke and D. E. Walker Celest. Mech P. C. Kammeyer Celest. Mech J. Casasayas and J. Llibre Mem. Am. Math. Soc M. Santoprete J. Math. Phys M. Corbera and J. Llibre Developments in Mathematical and Experimental Physics Vol. C: Hydrodynamics and Dynamical Systems ed. by Macias et al. Kluwer Academic Dordrecht New York 00 pp C. L. Siegel and J. K. Moser Lectures on Celestial Mechanics Springer-Verlag Berlin 1971.