NEW PERIODIC SOLUTIONS FOR NEWTONIAN n-BODY PROBLEMS WITH DIHEDRAL GROUP SYMMETRY AND 5 TOPOLOGICAL CONSTRAINTS 1 0 2 ZHIQIANGWANGANDSHIQINGZHANG r a M Abstract. Inthispaper,weprovetheexistenceofafamilyofnewnon- collisionperiodicsolutionsfortheclassicalNewtoniann-bodyproblems. 2 In our assumption, the n=2l ≥4 particles are invariant under the di- 1 hedralrotationgroupDl inR3suchthat,ateachinstant,thenparticles ] form two twisted l-regular polygons. Our approach is variational min- h p imizing method and we show that the minimizers are collision-free by - level estimates and local deformations. h at Keywords: Periodic Solutions, Newtonian n-body Problems, m Variational Method, DihedralGroup Symmetry,Topological [ Constraints. 2 v 1 4 1. Introduction and Main Result 7 7 0 Many authors (for example[6][8][9][18][22]) used the variational method . 1 to discover many new periodic solutions of the classical Newtonian n-body 0 5 problemsin thelast fifteen years. In particular, Chencinerand Montgomery 1 [6] proved the existence of the remarkable figure-8 type periodic solution for : v planar Newtonian 3-body problems with equal masses. Ferrario and Ter- i X racini [8] simplified and developed Marchal’s [12] important works and in- r a troduced the rotating circle property, proved that if the motion has certain symmetry under some group action having the rotation circle property, the solution exists and has no collision. Also Fusco et al.[9] proved the existence and collisionless of a number of new and interesting motions with the invariance of certain platonic polyhedra group action and some topological constraints. In this paper, we consider a system of n = 2l positive masses with their positions x(t) = (x (t),x (t),...,x (t))T moving in the space under 1 2 n 2010 Mathematics Subject Classification. 70F10,70F16,37C80,70G75. 1 2 ZHIQIANGWANGANDSHIQINGZHANG Newton’s law of gravitation: ∂U(x(t)) (1.1) x¨(t) = , M ∂x wherethepotentialfunctionU(x)= mimj and = diag m ,m ,...,m . i 0 . 0− { ∈ 0 · 3 } s { ∈ s · 3 } Now we state our main theorem: Theorem1.5. Forn = 2l 4andeveryintegers max 1,(n 1)3/2 π n ≥ ≤ { −n 23/2logn+γ− 1 where γ 0.57721566490153286 is the Euler-Mascheroni constant, there } ≈ exists a T-periodic solution u of the classical n-body problem. s ∗ ∈ K RUNNING HEAD 7 Corollary 1.6. For n = 2l 4 and s = 1, then h = l and there exists a ≥ T-periodic solution u of the classical Newtonian n-body problem. 1 ∗ ∈ K The case for n = 4 was discussed in Section 2 of [9], here we generalize theirresult. ByRemark1.3,scanbechosenlargerthan1whenn 8. Ifwe ≥ let f(n) = (n 1)3/2 π n 1, we see that f(n) is monotone increasing −n 23/2logn+γ − and we have f(4) 0.4697, f(6) 1.1400, f(8) 1.7376, f(10) 2.2931, ≈ ≈ ≈ ≈ f(14) 3.3262, f(26) 6.0995, etc. That is to say, for n 10, we can ≈ ≈ ≥ choose s = 2 such that there exists a T-periodic solution u of the 2 ∗ ∈ K classical n-body problem. Remark 1.7. Actually, the value of integer s depends on the estimate of excluding total collisions. By doing more explicit computation we can prove that, for n = 8, there exists a T-periodic solution u of the classical 2 ∗ ∈ K 8-body problem. 2. Coercivity Proposition 2.1. is coercive and ∂ . A|Ks Ks ⊂ △Gs Proof. From (1.2) we see that there is a collision if and only if there exists some t R such that u (t ) Γ, where Γ are the rotating axes of the c 0 c ∈ ∈ dihedral group D . So it is obvious that ∂ and in the following we l Ks ⊂ △Gs will prove the coercivity. Applied Newton-Leibniz Formula and Ho¨lder Inequality, we have T T 2h 2h u (T/2h) u (0) = u˙ (t)dt u˙ (t)dt 0 0 0 0 | − | | | ≤ | | Z0 Z0 . T ( T )12( 2h u˙0(t)2dt)12 ≤ 2h | | Z0 Since u (0) P ,u (T ) P+, we must have 0 ∈ 0− 0 2h ∈ s T T sπ u (0) u ( ) u (0) u ( ) cos , 0 0 0 0 · 2h ≤| || 2h | l 8 ZHIQIANGWANGANDSHIQINGZHANG which induces that T u˙ (t) 2 C u ( ) u (0)2 || 0 ||L2 ≥ 1| 0 2h − 0 | T T sπ C u ( )2+ u (0)2 2u (0) u ( ) cos 1 0 0 0 0 ≥ | 2h | | | − | || 2h | l h sπ T 2 siπ = C u (0) cos u ( ) +C u (0)2sin2 1 0 0 1 0 | | l −| 2h | | | l h i C u (0)2, 0 ≥ | | which implies that u˙ is an equivalent norm of H1 = W1,2 and the 0 L2 k k coercivity for the functional follows. (cid:3) A 3. Estimate on Collisions and the Proof of Theorem 1.5 In this section, we show that the minimizer u is free of collisions. s ∗ ∈ K Firstly we exclude total collisions and in Section 3.2 we discuss the partial collisions. 3.1. Total Collision. Our way to show that there is no total collision is based on level estimate. Firstly assuming that a total collision happens in u , we show that there is a lower bound of the action functional s ∗ ∈ K (u ), and we construct a test loop u˜ without collisions such that s B ≤ A ∗ ∈ K (u˜)< . Then we have (u˜) < (u ) which contradicts with that u is a A B A A ∗ ∗ minimizer. Thus the minimizers u is free of total collisions. s ∗ ∈ K Lemma3.1. (Gordon’s Theorem [10]) Letx W1,2([t ,t ],Rd)and x(t ) = 1 2 1 ∈ x(t ) = 0. Then for any a > 0, we have 2 t2(1 x˙ 2+ a )dt 3(2π)23a32(t2 t1)13. 2| | x ≥ 2 − Zt1 | | Proposition 3.2. (Lower bound estimates for with total collisions) A Assume that u has a total collision. Then s ∈ K (u) 3(n−1)(2h)23π23n32T13 , . A ≥ 4 B Proof. Since the center of mass is at origin, the functional can be written A as 1 T 1 M (3.1) (u) = m m [ u˙ u˙ 2+ ]dt, i j i j A M 2| − | u u i

New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints

Similar New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints