Hip Hop solutions of the N Boy poblem Esthe Baabés baabes@ima.ug.es Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Josep Maia Cos cos@eupm.upc.es Depatament e Matemàtica Aplicaa III, Univesitat Politècnica e Catalunya. Conxita Pinyol conxita.pinyol@uab.es Depatament Economia i Històia Econòmica, Univesitat Autònoma e Bacelona. Jaume Sole jaume.sole@ima.ug.es Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Abstact. Hip Hop solutions of the N boy poblem with equal masses ae shown to exist using an analytic continuation agument. These solutions ae close to plana egula N gon elative equilibia with small vetical oscillations. Fo fixe N, an infinity of these solutions ae thee imensional choeogaphies, with all the boies moving along the same close cuve in the inetial fame. Keywos: N-boy poblem, analytic continuation, Hip Hop, Choeogaphies. Intouction The equal mass n boy poblem has ecently attacte much attention thanks to the wok of Chencine an othe authos on the type of obits calle hip-hop solutions, an on the solutions that have eventually been calle choeogaphies. In a hip-hop solution, N boies of equal mass stay fo all time in the vetices of a egula otating anti-pism whose basis, i.e. the egula polygons that efine it, pefom an oscillatoy motion sepaating, eaching a maximum istance, appoaching, cossing each othe, an so on, as sketche in Figue fo N = 3. The N boies can be aange in two goups of N, each goup moving on its plane on a otating egula N gon configuation homogaphic, while the planes ae always pepenicula to the z-axis, oscillate along this axis, an coincie with opposite velocities at egula intevals when they coss the oigin. The othogonal pojection of both N gons on the z = 0 plane is always a egula otating N gon. On the othe han, a choeogaphy is a solution in which n boies move along the same close line in the inetial fame, chasing each othe at equi space intevals of time. It is well known the figue eight choeogaphy in the thee boy poblem, shown by Chencine c 006 Kluwe Acaemic Publishes. Pinte in the Nethelans. hiphop.tex; 6/0/006; 5:44; p.
Baabés, Cos, Pinyol, Sole an Montgomey 00 in a most celebate pape. A geat many choeogaphies with n > 3 have been shown numeically to exist by Simo 00. The above esults wee obtaine mostly by means of vaiational methos, which make it possible to fin solutions that o not epen on a small paamete, i.e. fa fom solutions of an integable poblem. See Chencine an Ventuelli, 000, Chencine et al., 00, Chencine an Féjoz, 005, an efeences theein fo etails. In the case of hip hop solutions, the question aises whethe in some simple cases they coul be obtaine though the taitional analytic continuation metho of Poincaé, which woul give families iffeentiable with espect to a paamete of peioic solutions, at least in a otating fame. In this espect, mention shoul be mae of a esult by Meye an Schmit 993 on a simila solution with a lage cental mass an n vey small, equal masses aoun it which was suggeste as a moel fo the baie stuctue of some of Satun s ings. In this pape we show that Poincae s agument of analytic continuation can be use to a vetical oscillations to the cicula motion of N boies of equal mass occupying the vetices of a egula N-gon. In this way, a family of thee-imensional obits, peioic in a otating fame, can be shown to exist. This is a Lyapunov family of obits whose peios ten to the peio of the vetical oscillations of the lineaize system aoun the elative equilibium solution. These solutions wee foun numeically by Davies et al. 983. Infinitely many of this solutions ae peioic in the inetial fame, povie that the quantity HN given by 7 oes not vanish, an ae thee imensional choeogaphies, in the sense that all boies move at equi space time intevals along a close twiste cuve in the inetial fame. Some solutions foun in ou aticle may coincie with the genealize hip hop solutions obtaine by Chencine in 00. Teacini an Ventuelli 005 ecently showe the existence of hip hop solutions in the same poblem using vaiational methos, aing vetical vaiations to the plana elative equilibium in oe to euce the value of the action functional. The vaiational appoach oes not epen on any small paamete an yiels global existence, while continuation methos give explicit appoximations to solutions in a small neighbouhoo of the elative equilibium. A pecise compaison of both methos fom a puely analytic point of view woul involve eithe estimating the istance fom the vaiational solutions to the elative equilibium o estimating the size of the neighbouhoo in which the family can be continue, but both questions seem fa fom easy. hiphop.tex; 6/0/006; 5:44; p.
Hip Hop solutions of the N Boy poblem 3 5 4 6 3 Figue. Qualitative epesentation of a Hip-Hop motion in the case of 6 boies. Equations of motion Consie N boies with equal mass m moving une thei mutual gavitational attaction an let i, ṙ i, i =,, N, be thei positions an velocities. The equations of motion of the N boy poblem ae i = Gm N,k i k i ki 3, whee ki = k i. Scaling the time t by Gm t the Lagangian function associate to the poblem becomes L = N i= ṙ i + i<j N i j. As we ae looking fo solutions of the N boy poblem such that all the boies stay fo all time on the vetices of an anti-pism, it will suffice to know the position an velocity of one of the N boies. Given = t, we efine i = R i, ṙ i = R i ṙ, fo i =,, N, whee R is a otation plus a eflection in such a way that all the boies ae on the vetices of an anti-pism. Since in this configuation N of the boies on a plane an the othe N on a paallel plane we can assume, without loss of geneality, that both planes ae pepenicula to the z axis. In this case, the matix R can be witten hiphop.tex; 6/0/006; 5:44; p.3
4 Baabés, Cos, Pinyol, Sole as cos π N sin π N 0 R = sin π N cos π N 0. 0 0 PROPOSITION.,,..., N =, R,..., R N is a solution of the N boy poblem given by if an only if t satisfies the equation N R k I = R k I 3. Poof. Substituting,,..., N =, R,..., R N in we have R i = N j=,j i R j R i R j R i 3 = N j=,j i R i R j i I R j i I 3, fo i =,..., N. Using the fact that R l = R N l, fo l =,..., N, we get = = = i j= i l= N R j i I R j i I 3 + N j=i+ R N l N i I R N l I 3 + R k I R k I 3. l= R j i I R j i I 3 R l I R l I 3 That is, we get the same equation fom the initial N equations. If = x, y, z is the position of the fist boy, then the equations of motion can be witten as the following iffeential system of oe two ẍ = Ux, y, z, ÿ = x Ux, y, z, z = y whee Ux, y, z is the potential function Ux, y, z = N Ux, y, z, 3 z. 4x + y sin kπ N + k z The poblem state by system 3 can be fomulate in Hamiltonian tems by the Hamiltonian function H = p x + p y + p z Ux, y, z, 4 hiphop.tex; 6/0/006; 5:44; p.4
Hip Hop solutions of the N Boy poblem 5 whee p x, p y an p z ae the momenta associate to the x, y, z cooinates. Intoucing cylinical cooinates by means of the canonical change x = cos θ, p x = p cos θ p θ sin θ, y = sin θ, p y = p sin θ + p θ cos θ, z =, p z = p, the Hamiltonian 4 becomes H = p + p θ + p N 4 sin kπ N Then the equations of motion fo the fist boy ae. 5 + k ṙ = p, θ = p θ, = p, p = p N θ 3 p θ = 0, p = N 4 sin kπ N sin kπ N + k 3/, k 4 sin kπ N + k 3/. 6 Since p θ = 0, the angula momentum p θ = Θ is constant an can be calculate fom the initial conitions. Then, once is obtaine, we will get θ fom the secon equation in 6. 3. Symmetic peioic solutions of the euce poblem We call euce poblem the poblem given by consieing in 6 only the equations fo the vaiables an, an complete poblem the whole set of equations 6. Ou aim is to fin peioic solutions of this euce poblem. This solutions will be, in geneal, quasi peioic solutions of the complete poblem. Consie the poblem pose by the Hamiltonian 5 fo a fixe value of the angula momentum p θ = Θ. The equations of motion of the hiphop.tex; 6/0/006; 5:44; p.5
6 Baabés, Cos, Pinyol, Sole euce poblem ae ṙ = p, = p, p = Θ N 3 p = N sin kπ N 4 sin kπ N + k 3/, k + k 3/, 4 sin kπ N which has a unique equilibium point,, p, p = a, 0, 0, 0, whee a = Θ /KN an is N KN = 4 sin kπ N The matix of the lineaize equations aoun this equilibium point whee an λ M = = 3Θ a 4 λ = 6a 3 S N = 6 0 0 0 0 0 0 λ 0 0 0 0 λ 0 0 + K N a 3 N = K8 N Θ 6, 7. 8, k = S N K6 N sin 3 kπ Θ 6, N N k. 9 sin 3 kπ N The matix M has two pais of imaginay eigenvalues ±iλ an ±iλ. By Lyapunov s cente theoem, Meye an Hall, 99, thee exist two one paamete families of peioic solutions, emanating fom the equilibium point povie that λ /λ is not a ational numbe. That is, it suffices to ensue that λ λ = 4 N N k sin 3 kπ N sin kπ N 0 hiphop.tex; 6/0/006; 5:44; p.6
Hip Hop solutions of the N Boy poblem 7 Table I. Values of λ /λ iffeent values of N N λ /λ fo 3 5 6 44 7 5 5 + 4 3 60 + 5 0 + 5 + 4 + 5 + 6 5 + 6 + 4 3 + 6 is not the squae of a ational numbe. A numbe of values of this expession fo iffeent values of N ae shown on Table I. The = p = 0 plane is invaiant an the -moe solutions lie in this plane with peios appoaching π/λ. These ae actually the homogaphic solutions nea the elative equilibium. The -moe solutions ae thee imensional obits whose peios ten to π/λ an this is the family we ae inteeste in. The equations of motion of the euce poblem 7 ae invaiant by the symmeties S : t,,, p, p t,,, p, p, S : t,,, p, p t,,, p, p, an we have the following well known poposition. PROPOSITION. Let qt = t, t, p t, p t be a solution of the equations 7. If qt satisfies that 0, p 0 = 0, 0 an p T, p T = 0, 0, then qt is a oubly symmetic peioic solution of peio 4T. We will show the existence of oubly symmetic peioic obits in system 7. Let q 0 = 0, 0, p 0, p 0 = 0, 0, p 0, 0. The solution of system 7 with these initial conitions is given by t = 0, fo all t, togethe with any solution t of the Keple poblem = Θ 3 K N, with K N given by 8. As is well known, its solutions can be witten as t = a e cos Et, hiphop.tex; 6/0/006; 5:44; p.7
8 Baabés, Cos, Pinyol, Sole whee a is the semimajo axis, e the eccenticity an E the eccentic anomaly. The function t is peioic of peio T = πa 3/ /K N = π/λ an a e K N = Θ. These solutions will be calle plana as oppose to the spatial o thee imensional solutions when t the istance fom the fist boy to the z = 0 plane is not ientically zeo. In oe to obtain peioic solutions of the euce poblem, we a petubations to peioic plana obits in the vetical iection. If the petubation is small enough, the motion can be ecouple into a plana plus a vetical motion in a fist appoximation. Substituting by ε on the equations 7, an keeping tems in ɛ, we obtain whee ṙ = p, = p, ṗ ṗ = Θ 3 K N + 3S N 4 ɛ + Oɛ 4, = S N 3 + 3W N 5 3 ɛ + Oɛ 4, WN = N k 4 64. sin 5 kπ N System can then be witten as whee q = Fq,ɛ = F 0 q + ɛ F q + Oɛ 4 3 F 0 q = p, p, Θ 3 K N, S N 3, F q = 0, 0, 3S N 4, 3W N 5 3. Let q 0 be a vecto of initial conitions. The solution of 3 with initial value q 0 at t = 0 can be expane as a powe seies in ɛ as qt, q 0, ɛ = q 0 t, q 0 + ɛ q t, q 0 + Oɛ 4, whee q 0 t, q 0 is the solution of the unpetube poblem q 0 t, q 0 = F 0 q 0 4 with initial conitions q 0, an q t, q 0 is the solution of q t, q 0 = F q 0 t, q 0 + DF 0 q 0 t, q 0 q t, q 0 with initial conitions q 0, q 0 = 0. The enties of the matix DF ae the patial eivatives of F with espect to the q vaiable, an by the fomula of Lagange we have hiphop.tex; 6/0/006; 5:44; p.8
Hip Hop solutions of the N Boy poblem 9 whee t q t, q 0 = Qt, q 0 Q τ, q 0 F q 0 τ, q 0 τ, 5 0 Qt, q 0 = q0 t, ξ ξ 6 ξ=q0 THEOREM. Let T0 = k+ π λ, a = Θ an q KN 0 = a, 0, 0, p 0. Assume that λ /λ given by 0 is not a ational numbe. Then thee exist 0, T such that the solution qt, q 0, ɛ of system with initial conitions q 0 = a+ 0, 0, 0, p 0 is a oubly symmetic peioic solution of peio 4T0 + T. The functions 0 an T ae given by 0 = ɛ 3 p 0 Sn a 4 λ λ 4 λ + Oɛ 4 T = ɛ 9 p 0 a 5 λ 3 λ 4 λ whee an B k, N + 5 3 λ B k, N + Oɛ 4 B k, N = S N λ λ sin + k π λ λ, B k, N = + k π W N λ 3 λ 4 λ + SN λ B, 7 B k, N = λ π+kλ 4λ 3λ 8λ +3λ 3 λ λ sin + kπ λ. λ Note that fo ɛ 0 the peios of the solutions given by the theoem ten to π/λ an they belong to a symmetic Lyapunov family. Poof. Notice that the solution of the unpetube poblem 4 with initial conition q 0 is This solution satisfies q 0 t, q 0 = a, p 0 λ sinλ t, 0, p 0 cosλ t. 8 q 0 T 0, q 0 = a, p 0 k λ, 0, 0, 9 an q 0 t, q 0 is a oubly symmetic peioic solution of the unpetube system of peio 4T0. hiphop.tex; 6/0/006; 5:44; p.9
0 Baabés, Cos, Pinyol, Sole We must fin initial conitions q 0 = a + 0, 0, 0, p 0 an T = T0 + T such that the solution qt, q 0, ɛ of system satisfies { p T0 + T, q 0, ɛ = p 0 T0 + T, q 0 + ɛ p T0 + T, q 0 + Oɛ 4 = 0 p T 0 + T, q 0, ɛ = p 0 T 0 + T, q 0 + ɛ p T 0 + T, q 0 + Oɛ 4 = 0 0 By Poposition, qt, q 0, ɛ will be a oubly symmetic peioic solution of peio 4T0 + T. Fo a fixe value p 0, an k = 0,, we consie p k T0 + T, q 0 an p k T 0 + T, q 0 as functions of T, 0. Expaning 0 as powe seies in the s we get 0 p 0 T0 =, q 0 0 p 0 T + 0, q 0 +ɛ p T0, q 0 p T + 0, q 0 +Oɛ 4 p 0 p 0 p 0 p 0 p p T 0,q 0 p p T 0 + O T, 0 + T 0,q 0 T 0 + O T, 0 Now we have fom 9 that 0 p T0, q 0 0 p 0 T 0, q 0 = 0 p 0 p 0 so that if p 0 p 0 0, the system 0 can be solve fo T 0,q 0 T, 0 in a neigbouhoo of 0, 0 by means of the implicit function theoem. An appoximation to T, 0 can be easily compute fom p 0 T = ɛ p 0 p T0, q 0 0 p T + O T, 0 +Oɛ 4 0, q 0 p 0 The functions p0 the tems p0 p 0 p 0 p 0 p 0 an p T0, q 0, p0 T 0,q 0 =, p p 0, p0 T 0,q 0 can be compute fom 8. In oe to get we must compute 6. Then 0 λ sin k+ λ λ π p 0λ k 3 p 0 λ k +k π λ sin λ λ λ a λ λ 4 λ T 0, q 0 ae the last two components of q T 0, q 0 = QT 0, q 0 T 0 0 Q τ, q 0 F q 0 τ, q 0 τ, hiphop.tex; 6/0/006; 5:44; p.0
Hip Hop solutions of the N Boy poblem Table II. Numeical values of T, 0 fo Θ =, N = 3 an k = 0. p 0 = p 0 = p 0 = 0.5 ɛ = 0.03 0.000660, 0.000496 0.00064, 0.0003 0.00004, 0.000030 ɛ = 0.06 0.0066, 0.00997 0.000660, 0.000496 0.00064, 0.0003 ɛ = 0.09 0.00606, 0.004534 0.00490, 0.009 0.00037, 0.00079 ɛ = 0. 0.00963, 0.00866 0.0066, 0.00997 0.000660, 0.000496 which can be easily compute an ae given by p z 0 = p z 0 = 3 p 0 S N a 4 λ λ 4 λ + kπ sin 9 k p 3 0 3 a 5 λ 3 λ 4 λ 4 λ B, k, N λ, 3 λ whee B k, N is given by 7. Finally we can substitute 3 an in an we obtain the appoximation to T, 0. Theoem gives an appoximation to initial conitions fo oubly symmetic peioic obits fo ɛ sufficiently small. This esults have been checke numeically an a goo ageement has been obtaine. Fo fixe values of N, k, Θ, p 0 an ɛ, we compute numeically the values T, nea to T0, a such that the obit with initial conitions q =, 0, 0, p 0 is a oubly symmetic peioic obit of peio 4T. The integation of the iffeential equations has been one by means of a Runge-Kutta RK78 algoithm. Then we compute T = T T0 an 0 = a. Table II shows the numeically compute values of T, 0, fo Θ = an iffeent values of ɛ. hiphop.tex; 6/0/006; 5:44; p.
Baabés, Cos, Pinyol, Sole 4. Hip Hop peioic obits an choeogaphies The question whethe obits which ae peioic in the euce system ae peioic also in the inetial fame is of couse only a question of commensuability between π an the angle otate in the inetial system in a peio. If this angle can be seen to change along the family of peioic solutions, then thee will exist infinitely many peioic solutions in the inetial system wheneve its value is commensuable with π. It suffices then to see that its eivative with espect to ɛ is iffeent fom zeo fo ɛ = 0. Now, a small vaiation on this simple agument shows that thee exist infinitely many choeogaphies as well. Think of the obit as having peio 4T, whee T is the time spent fom the initial plana position as a egula N gon to the maximum sepaation of the planes containing the N gon configuations. Afte a time T, the N boies that at t = 0 whee thown upwas will hit the initial plane with a velocity symmetic to the initial one, which is exactly the initial velocity of the othe N boies, which wee thown ownwas. If at t = T the position of N boies is the same as the position at t = 0 of the othe N boies, they will follow the same path, so we have the kin of motion that has been calle a choeogaphy. We give an outline of the computation of the eivative. As we have seen in the pevious Section, fo small values of ɛ we can obtain peioic solutions qt, q 0, ɛ of the euce poblem fo initial conitions q 0 = a + 0, 0, 0,p 0 an peio 4T = 4T 0 + 4 T. Fo a fixe value p 0, the function t is given by t, q 0, ɛ = 0 0 t, a + 0 + O 0 + q 0 +ɛ t, a + 0 + O 0 + Oɛ 4 q 0 = a + cosλ t 0 + ɛ a, t + Oɛ 4 4 whee 0 is given in Theoem, an a, t = 3 p 0 S N sint λ λ + + cost λ λ a 4 λ λ 4 4 λ λ is the fist component of the vecto q t as given by 5. hiphop.tex; 6/0/006; 5:44; p.
Hip Hop solutions of the N Boy poblem 3 Thus, in oe to fin peioic solutions in the inetial efeence system, fo a fixe Θ we must fin solutions of the equation with θ0 = 0 at such that θ = Θ 5 θ4qt = πp 6 fo some integes p an q. Substituting 4 in 5, we get that θ = Θ a cosλ t 0 + ɛ a, t + Oɛ 4 a Integating an emembeing that T is Oɛ, we have θ4t = Θ a 4T 0 + ɛ 3 + kπp 0 Θ 6 whee 8 a 7 K N 7 S N 5 K N 4 S N HN + Oɛ4 HN = 8 S N 6 + 3 K N 4 W N + K N S N 4 S N W N 7 an K N, S N, W N ae efine, espectively, in 8, 9 an. Thus, it is enough to see that the tem HN is iffeent fom zeo to guaantee that thee exist infinitely many values of the paamete ɛ such that 6 hols. This is inee the case fo N 0, an pobably fo infinitely many values of N, although we o not have a fomal poof of this fact. Acknowlegements The fist autho is patially suppote by DGES gant BFM003-09504- C0-0. The secon an fouth authos ae patially suppote by DGES gant numbe BFM00-0436-C0-0 an by a DURSI gant numbe 00SGR 0073. The thi autho is patially suppote by CI- CYT gant numbe SEC003-05/ECO an by DURSI gant numbe SGR00-64. Refeences Chencine, A.: 00, Action minimizing solutions of the newtonian n-boy poblem: fom homology to symmety. Poceeings of the Intenational Congess of Mathematicians Vol. III, Beijing, 00, 79 94. hiphop.tex; 6/0/006; 5:44; p.3
4 Baabés, Cos, Pinyol, Sole Chencine, A. an Féjoz, J.: 005, L équation aux vaiations veticales un équilibe elatif comme souce e nouvelles solutions péioiques u poblème es N cops, C. R. Math. Aca. Sci. Pais 3408, 593 598. Chencine, A., Geve, J., Montgomey, R. an Simó, C.: 00, Simple choeogaphic motions of N boies: a peliminay stuy, Geomety, mechanics, an ynamics, Spinge, New Yok, pp. 87 308. Chencine, A. an Montgomey, R.: 000, A emakable peioic solution of the thee-boy poblem in the case of equal masses, Ann. of Math. 53, 88 90. Chencine, A. an Ventuelli, A.: 000, Minima e l intégale action u poblème newtonien e 4 cops e masses égales ans R 3 : obites hip-hop, Celestial Mech. Dynam. Astonom. 77, 39 5 00. Davies, I., Tuman, A. an Williams, D.: 983, Classical peioic solution of the equal-mass n-boy poblem, n-ion poblem an the n-electon atom poblem, Phys. Lett. A 99, 5 8. Meye, K. R. an Hall, G. R.: 99, Intouction to Hamiltonian ynamical systems an the N-boy poblem, Vol. 90 of Applie Mathematical Sciences, Spinge- Velag, New Yok. Meye, K. R. an Schmit, D. S.: 993, Libations of cental configuations an baie Satun ings, Celestial Mech. Dynam. Astonom. 553, 89 303. Simó, C.: 00, New families of solutions in N boy poblems, Poceeings of the thi Euopean Congess of Mathematics, in Pog. Math. Vol. 0, 0 5. Teacini, S. an Ventuelli, A.: 005, Symmetic tajectoies fo the n-boy poblem with equal masses, pepint. hiphop.tex; 6/0/006; 5:44; p.4