New Periodic Solutions for Some Planar N+3 -Body Problems with Newtonian Potentials
aa r X i v : . [ m a t h - ph ] N ov New Periodic Solutions for Some Planar N + 3-Body Problems with Newtonian Potentials ∗ Pengfei Yuan and Shiqing [email protected], [email protected]
Department of Mathematics, Sichuan University, Chengdu 610064, China
Abstract
For some planar Newtonian N +3-body problems, we use variational minimizationmethods to prove the existence of new periodic solutions satisfying that N bodieschase each other on a curve, and the other 3 bodies chase each other on anothercurve. From the definition of the group action in equations (3 . − (3 . Key Words: N +3-body problems, periodic solutions, winding numbers, variationalminimizers. In recent years, many authors used methods of minimizing the Lagrangian action on asymmetric space to study the periodic solutions for Newtonian N -body problem ([2] , [4] − [6] , [8] − [29] , [31] − [40]). Especially, A.Chenciner-R.Montgomery [16] proved the existenceof the remarkable figure eight type periodic solution for Newtonian three-body problemwith equal masses, C.Sim´o [32] discovered many new periodic solutions for Newtonian N -body problem using numerical methods. C.Machal [27] studied the fixed-ends (Bolza)problem for Newtonian N -body problem and proved that the minimizer for the Lagrangianaction has no interior collision; A.Chenciner [12], D.Ferario and S.Terracini [22] simplifiedand developed C.Marchal’s important works; S.Q.Zhang [36], S.Q.Zhang, Q.Zhou ([37] − [40]) decomposed the Lagrangian action for N -body problem into some sum for two-bodyproblem and compared the lower bound for the lagrangian action on test orbits with theupper bound on collision set to avoid collisions under some cases. Motivated by the works ofA.Chenciner and R.Montgomery, C.Sim´o, C.Marchal, S.Q.Zhang and Q.Zhou, K.C. Chen ∗ Supported partially by NSF of China. − [11]) studied some planar N -body problems and got some new planar non-collisionperiodic and quasi-periodic solutions.The equations for the motion of the Newtonian N -body problem are: m i ¨ q i = ∂U ( q ) ∂q i , i = 1 , . . . , N, (1.1)where q i ∈ R k denotes the position of m i , and the potential function is : U = X ≤ i Let Γ : x ( t ) , t ∈ [ a, b ] be a given oriented continuous closed curve,and p a point of the plane, not on the curve. Then the mapping ϕ : Γ → S given by ϕ ( x ( t )) = x ( t ) − p | x ( t ) − p | , t ∈ [ a, b ] , (1 . is defined to be the position mapping of the curve Γ relative to p . When the point on Γ goesaround the given oriented curve once, its image point ϕ ( x ) will go around S in the samedirection with Γ a number of times. When moving counter-clockwise or clockwise, we setthe sign + or − , and we denote it by deg (Γ , p ) . If p is the origin, we denote it by deg (Γ) . C.H.Deng and S.Q.Zhang [20], X.Su and S.Q.Zhang [33] studies periodic solutions fora class of planar N + 2-body problems, they defined the following orbit spaces:Λ = { q ∈ E | q i ( t + Tr ) = O ( 2 πr ) q i ( t ) , i = 1 , . . . , N + 2; q i +1 ( t ) = q i ( t + TN ) , i = 1 , . . . , N − , q ( t ) = q N ( t + TN ); q i ( t + TN ) = q i ( t ) , i = N + 1 , N + 2 , ∀ t > } (1.6)2nd Λ = { q ∈ Λ | q i ( t ) = q j ( t ) , ∀ i = j, ∀ t ∈ R ; deg ( q i ( t ) − q j ( t )) = 1 , ≤ i = j ≤ N, deg ( q N +1 ( t ) − q N +2 ( t )) = k } , (1.7)where E = { q = ( q , q , . . . , q N +2 ) | q i ( t ) ∈ W , ( R /T Z , R ) , N +2 X i =1 m i q i ( t ) = 0 } , (1 . O ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . Motivated by their work, we consider N + 3-body problems( N > N and 3 are coprime),the equations of the motion are: m i ¨ q i ( t ) = ∂U ( q ) ∂q i , i = 1 , . . . , N + 3 . (1 . = { q ∈ E | q i ( t + Tr ) = O ( 2 πdr ) q i ( t ) , i = 1 , . . . , N + 3; q i +1 ( t ) = q i ( t + TN ) , i = 1 , . . . , N, q ( t ) = q N ( t + TN ); q N + j ( t ) = q N + j − ( t + T , j = 2 , , q N +1 ( t ) = q N +3 ( t + T q i ( t + T q i ( t ) , i = 1 , . . . , N ; q j ( t + TN ) = q j ( t ) , j = N + 1 , N + 2 , N + 3 } , (1.10)and Λ = { q ∈ Λ | q i ( t ) = q j ( t ) , ∀ i = j, ∀ t ∈ R ; deg ( q i ( t ) − q j ( t )) = k , ≤ i < j ≤ N ; deg ( q i ′ ( t ) − q j ′ ( t )) = k , N + 1 ≤ i ′ < j ′ ≤ N + 3 } , where E = { q = ( q , q , . . . , q N +3 ) | q i ( t ) ∈ W , ( R /T Z , R ) , N +3 X i =1 m i q i ( t ) = 0 } . (1 . r, k , k , d satisfy the following compatible conditions: k = d ( mod r ) , k = d ( mod r ) , k = 3 s , k = N s , s , s ∈ Z . (1 . N and 3 are coprime, we have ( N, 3) = 1. In this paper, we also require r and 3coprime, so ( r, 3) = 1.We get the following theorem: Theorem 1.1 (1) Consider the seven-body problems (1 . of equal masses, for r =7 , k = 3 , k = − , d = 3 , then the global minimizer of f on ¯Λ is a non-collision periodicsolution of (1 . . (2) Consider the eight-body problems (1 . of equal masses, for r = 8 , k = 3 , k = − , d = 3 , then the global minimizer of f on ¯Λ is a non-collision periodic solution of (1 . . (3) Consider the ten-body problems (1 . of equal masses, for r = 10 , k = 3 , k = − , d = 3 , then the global minimizer of f on ¯Λ is a non-collision periodic solution of (1 . . Lemma 2.1. (Eberlein-Shmulyan [7] ) A Banach space X is reflexive if and only if anybounded sequence in X has a weakly convergent subsequence. Lemma 2.2. ([7]) Let X be a real reflexive Banach space, M ⊂ X is a weakly closedsubset, f : M → R is weakly semi-continuous.If f is coercive, that is, f ( x ) → + ∞ as k x k→ + ∞ , then f ( x ) attains its infimum on M . Lemma 2.3. ([30]) Let G be a group acting orthogonally on a Hilbert space H . Definethe fixed point space F G = { x ∈ H | g · x = x, ∀ g ∈ G } , if f ∈ C ( H, R ) and satisfies f ( g · x ) = f ( x ) for any g ∈ G and x ∈ H , then the critical point of f restricted on F G isalso a critical point of f on H . Lemma 2.4. ([41]) Let q ∈ W , ( R /T Z , R n ) and R T q ( t ) d t = 0 , then we have ( i ) . Poincare-Wirtinger’s inequality: Z T | ˙ q ( t ) | d t ≥ (cid:16) πT (cid:17) Z T | q ( t ) | d t. (2 . ii ) . Sobolev’s inequality: max ≤ t ≤ T | q ( t ) | = k q k ∞ ≤ r T (cid:0) Z T | ˙ q ( t ) | d t (cid:1) / . (2 . Lemma 2.5. (Gordon[24])(1) Let x ( t ) ∈ W , ([ t , t ] , R k ) and x ( t ) = x ( t ) = 0 , Then forany a > , we have Z t t ( 12 | ˙ x | + a | x | ) dt ≥ 32 (2 π ) a ( t − t ) . (2 . (2)(Long and Zhang [26] ) Let x ( t ) ∈ W , ( R/T Z, R k ) , R T xdt = 0 , then for any a > , wehave Z T ( 12 | ˙ x ( t ) | + a | x | ) dt ≥ 32 (2 π ) a T . (2 . Proof of Theorem 1.1 we consider the system (1 . 9) of equal masses. Without loss of generality, we suppose thatthe masses m = m = · · · = m N +3 = 1, and the period T = 1.Define G = Z r × Z × Z N and the group action g = h g i × h g i × h g i on the space E : g ( q ( t ) , . . . , q N +3 ( t )) = ( O ( − πdr ) q ( t + 1 r ) , . . . , O ( − πdr ) q N +3 ( t + 1 r )) (3.1) g ( q ( t ) , . . . , q N +3 ( t ))= ( q ( t + 13 ) , . . . , q N ( t + 13 ) , q N +3 ( t + 13 ) , q N +1 ( t + 13 ) , q N +2 ( t + 13 )) (3.2) g ( q ( t ) , . . . , q N +3 ( t ))= ( q N ( t + 1 N ) , q ( t + 1 N ) . . . , q N − ( t + 1 N ) , q N +1 ( t + 1 N ) , q N +2 ( t + 1 N ) , q N +3 ( t + 1 N ))(3.3)This implies that Λ is the fixed point space of g on E . Furthermore, for any g i and q ∈ E , we have f ( g i · q ) = f ( q ) for i = 1 , , 3. Then the Palais symmetry principle impliesthat the critical point of f restricted on Λ is also a critical point of f on E . Lemma 3.1. The critical point of minimizing the Lagrangian functional f restricted on Λ (with winding number restriction) is also a critical point of f on Λ , then it is also thesolution of (1 . . The proof is similar to that of Lemma 3 . q i ( t ) = O ( − πdr ) q i ( t + 1 r )( i = 1 , · · · , N + 3) , we have Z q i ( t ) dt = 0 . Then the Lemma 2 . Z | ˙ q i ( t ) | dt ≥ (2 π ) Z | q i ( t ) | dt. Hence f ( q ) is coercive on ¯Λ . It is easy to see that ¯Λ is a weakly closed subset.Fatou’slemma implies that f ( q ) is a weakly lower semi-continuous. Then by Lemma 2 . f ( q )attains inf { f ( q ) | q ∈ ¯Λ } . Similar to Lemma 3 . Lemma 3.2. The limit curve q ( t ) = ( q ( t ) , q ( t ) , . . . , q N +3 ( t )) ∈ ∂ Λ of a sequence q l ( t ) =( q l ( t ) , q l ( t ) , . . . , q lN +3 ( t )) ∈ Λ may either have collisions between some two point massesor has the same winding number ( i.e.deg ( q i ( t ) − q j ( t )) = k , ≤ i = j ≤ N ; deg ( q i ′ ( t ) − q j ′ ( t )) = k , N + 1 ≤ i ′ = j ′ ≤ N + 3) . 5n the following, we prove that the minimizer of f is a non-collision solutions of thesystem (1 . . Since P N +3 i =1 q i = 0, by the Lagrangian identity, we have f ( q ) = 1 N + 3 X ≤ i 4) is a Lagrangian action for a suitabletwo body problem, which is a key step for the lower bound estimate on the collision set.We estimate the infimum of the action functional on the collision set. Since the sym-metry for a two-body problem implies that the Lagrangian action on a collision solutionis greater than that on the non-collision solution, and the more collisions there are, thegreater the Lagrangian is. We only assume that the two bodies collide at some moment t , without loss of generality, let t = 0, we will sufficiently use the symmetries of collisionorbits.since q ∈ ¯Λ , we have q i ( t + 1 r ) = O ( 2 πdr ) q i ( t ) , i = 1 , . . . , N + 3; (3.5) q i +1 ( t ) = q i ( t + 1 N ) , i = 1 , . . . , N − , q ( t ) = q N ( t + 1 N ); (3.6) q N +2 ( t ) = q N +1 ( t + 13 ) , q N +3 ( t ) = q N +2 ( t + 13 ) , q N +1 ( t ) = q N +3 ( t + 13 ); (3.7) q i ( t + 13 ) = q i ( t ) , i = 1 , . . . , N ; (3.8) q j ( t + 1 N ) = q j ( t ) , j = N + 1 , N + 2 , N + 3 . (3.9)Case 1: q , q collide at t = 0.By (3 . q , q collide at t = ir , i = 0 , . . . , r − . Furthermore, by (3 . q , q collide at t = ir , ir + 13 , ir + 23 ( mod . (3 . . 6) and (3 . q , q collide at ir + N − N , ir + 13 + N − N , ir + 23 + N − N ( mod , i = 0 , . . . , r − ,q , q collide at ir + N − N , ir + 13 + N − N , ir + 23 + N − N ( mod , i = 0 , . . . , r − , q N − , q N collide at ir + 2 N , ir + 13 + 2 N , ir + 23 + 2 N ( mod , i = 0 , . . . , r − ,q N , q collide at ir + 1 N , ir + 13 + 1 N , ir + 23 + 1 N ( mod , i = 0 , . . . , r − . Lemma 3.3. ∀ ≤ i, j ≤ r − , ≤ k ≤ , ( i − j ) + k = 0 , we have ir = jr + k mod 1) (3 . Proof. If there exist 0 ≤ i , j ≤ r − , ≤ k ≤ , ( i − j ) + k = 0 such that i r = j r + k mod . Then we have 1 | ( j r + k − i r ) . Since j r + k − i r ≥ − r − r = − r > − , and j r + k − i r ≤ r − r + 23 < , we can deduce j r + k − i r = 0 or j r + k − i r = 1 . If j r + k − i r = 0, then 3( i − j ) = k r . When k = 0, we get i = j , which is acontradiction with our assumptions on the i , j , k ; when k = 0, notice 0 < k ≤ 2, wecan deduce 3 | r , which is a contradiction since ( r, 3) = 1 . If j r + k − i r = 1, then 3( j − i ) = (3 − k ) r . When k = 0, we get r = j − i , whichis a contradiction since − r + 1 ≤ j − i ≤ r − 1; when k = 0, notice 1 ≤ − k < 3, wecan deduce 3 | r , which is also a contradiction since ( r, 3) = 1 . By (3 . 10) and Lemma 3 . 3, we know that q , q collide at t i = i r , i = 0 , . . . , r − . (3 . . 5, (3 . Z ( 12 | ˙ q ( t ) − ˙ q ( t ) | + N + 3 | q ( t ) − q ( t ) | ) dt = r − X i =0 Z t i +1 t i ( 12 | ˙ q ( t ) − ˙ q ( t ) | + N + 3 | q ( t ) − q ( t ) | ) dt ≥ × (2 π ) ( N + 3) r ( 13 r ) . (3.13)7rom (3 . 6) and (3 . q , q collide at i r + N − N ( mod , i = 0 , . . . , r − ,q , q collide at i r + N − N ( mod , i = 0 , . . . , r − , ... q N − , q N collide at i r + 2 N ( mod , i = 0 , . . . , r − , (3.14) q N , q collide at i r + 1 N ( mod , i = 0 , . . . , r − . (3.15) Lemma 3.4. ∀ ≤ i, i ′ ≤ r − , ≤ j, j ′ ≤ N − , ( i − i ′ ) + ( j − j ′ ) = 0 , we have i r + jN = i ′ r + j ′ N ( mod . (3 . . Remark 3.1 From Lemma 3 . ∀ ≤ i, i ′ ≤ r − , ≤ j, j ′ ≤ N − , ≤ k, k ′ ≤ , ( i − i ′ ) + ( j − j ′ ) + ( k − k ′ ) = 0, we have ir + jN + k = i ′ r + j ′ N + k ′ mod . By Lemma 2 . 5, Lemma 3 . . Z ( 12 | ˙ q j +1 ( t ) − ˙ q j +2 ( t ) | + N + 3 | q j +1 ( t ) − q j +2 ( t ) | ) dt ≥ × (2 π ) ( N + 3) r ( 13 r ) , ( j = 1 , . . . , N − , (3.17) Z ( 12 | ˙ q N ( t ) − ˙ q ( t ) | + N + 3 | q N ( t ) − q ( t ) | ) dt ≥ × (2 π ) ( N + 3) r ( 13 r ) . (3.18)Let M = N − X j =0 Z ( 12 | ˙ q j +1 ( t ) − ˙ q j +2 ( t ) | + N + 3 | q j +1 ( t ) − q j +2 ( t ) | ) dt + Z ( 12 | ˙ q N ( t ) − ˙ q ( t ) | + N + 3 | q N ( t ) − q ( t ) | ) dt. . , (3 . , (3 . . 5, and notice that ∀ ≤ i ≤ N, N + 1 ≤ j ≤ N + 3 , R q i ( t ) dt = 0 , R N q j ( t ) dt = 0, so we have f ( q ) = 1 N + 3 X ≤ i 2) collide at t = 0.By (3 . q , q k +2 ( k = 1 , . . . , N − 2) collide at t = ir , i = 0 , . . . , r − . Then by (3 . 8) , q , q k +2 collide at t = ir , ir + 13 , ir + 23 ( mod , i = 0 , · · · , r − . (3 . . 3, we get q , q k +2 collide at t = i r , i = 0 , . . . , r − . (3 . . q , q k +3 collide at t = i r + N − N ( mod , i = 0 , . . . , r − ,q , q k +4 collide at t = i r + N − N ( mod , i = 0 , . . . , r − , ... q N − k − , q N collide at t = i r + k + 2 N ( mod , i = 0 , . . . , r − ,q N − k , q , collide at t = i r + k + 1 N ( mod , i = 0 , . . . , r − ,q N − k +1 , q collide at t = i r + kN ( mod , i = 0 , . . . , r − , q N , q k +1 collide at t = i r + 1 N ( mod , i = 0 , . . . , r − . (3.22)Then by Lemma 2 . 5, Lemma 3 . 3, Lemma 3 . 4, (3 . − (3 . f ( q ) ≥ × ( 4 π N + 3 ) [ N × r ( 13 r ) + 3 × ( 13 ) ( C N − N ) + 3 N + 3 N ( 1 N ) ]= A. (3.23)Case 3: q , q N +1 collide at t = 0.By (3 . , (3 . , (3 . q , q N +1 collide at t = ir , ir + 13 , ir + 23 ,ir + N N , ir + 13 + N N , ir + 23 + N N ( mod , i = 0 , . . . , r − . (3.24)Simplify (3 . 24) , we get q , q N +1 collide at t = ir + j , i = 0 , . . . , r − , j = 0 , . . . , Lemma 3.5. ∀ ≤ i, i ′ ≤ r − , ≤ j, j ′ ≤ , ( i − i ′ ) + ( j − j ′ ) = 0 , we have ir + j = i ′ r + j ′ mod 1) (3 . Proof. If there exist 0 ≤ i , i ≤ r − , ≤ j , j ≤ , ( i − i ) + ( j − j ) = 0 such that i r + j i r + j mod 1) (3.27)Since i r + j − i r − j ≥ − r − r − > − ,i r + j − i r − j ≤ r − r + 56 < , then we deduce i r + j − i r − j − i r + j − i r − j i r + j − i r − j i r + j − i r − j − 1, we have r (6 + j − j ) = 6( i − i ). When i = i , which is acontradiction since r (6 + j − j ) = 0 ; when i = i and j = j , we can deduce r = i − i ,which is a contradiction since − r + 1 ≤ i − i ≤ r − 1; when i = i and j = j , we candeduce 6 | r , which is a contradiction since ( r, 3) = 1.We can use similar arguments to prove i r + j − i r − j = 0 and i r + j − i r − j = 1.10rom (3 . 25) and (3 . q , q N +1 collide at t i = i r , r = 0 , . . . , r − . (3 . . . Z ( 12 | ˙ q ( t ) − ˙ q N +1 ( t ) | + N + 3 | q ( t ) − q N +1 ( t ) | ) dt = r − X i =0 Z t i +1 t i ( 12 | ˙ q ( t ) − ˙ q N +1 ( t ) | + N + 3 | q ( t ) − q N +1 ( t ) | ) dt ≥ × (2 π ) ( N + 3) r ( 16 r ) . (3.29)By (3 . . q , q N +2 , collide at t = i r + N − N , i = 0 , . . . , r − ,q , q N +3 , collide at t = i r + N − N , i = 0 , . . . , r − , ... q N , q N collide at t = i r + 1 N , i = 0 , . . . , r − . (3.30) Lemma 3.6. ∀ ≤ i, i ′ ≤ r − , ≤ j, j ′ ≤ N − , ( i − i ′ ) + ( j − j ′ ) = 0 , we have i r + jN = i ′ r + j ′ N . (3 . . . 5, Lemma 3 . 6, (3 . − (3 . Z ( 12 | ˙ q j +1 ( t ) − ˙ q N + j +1 ( t ) | + N + 3 | q j +1 ( t ) − q N + j +1 ( t ) | ) dt ≥ × (2 π ) ( N + 3) r ( 16 r ) ( j = 1 , . . . , N − . (3.32)Let M = N − X j =0 Z ( 12 | ˙ q j +1 ( t ) − ˙ q N + j +1 ( t ) | + N + 3 | q j +1 ( t ) − q N + j +1 ( t ) | ) dt . 5, Lemma 3 . 6, (3 . 29) and (3 . f ( q ) = 1 N + 3 X ≤ i 2) collide at t = 0.By (3 . , (3 . q , q k +2 ( k = 1 , . . . , N +12 − 2) collide at t = ir , ir + 13 , ir + 23 ( mod , i = 0 , . . . , r − , (3 . . 3, we get q , q k +2 ( k = 1 , . . . , N +12 − 2) collide at t = i r , i = 0 , . . . , r − , (3 . . q , q k +3 collide at t = i r + N − N ( mod , i = 0 , . . . , r − ,q , q k +4 collide at t = i r + N − N ( mod , i = 0 , . . . , r − , ... q N − k − , q N collide at t = i r + k + 2 N ( mod , i = 0 , . . . , r − ,q N − k , q , collide at t = i r + k + 1 N ( mod , i = 0 , . . . , r − ,q N − k +1 , q collide at t = i r + kN ( mod , i = 0 , . . . , r − , ... q N , q k +1 collide at t = i r + 1 N ( mod , i = 0 , . . . , r − . (3.36)12hen by Lemma 2 . 5, Lemma 3 . 4, (3 . , (3 . f ( q ) ≥ × ( 4 π N + 3 ) [ N × r ( 13 r ) + 3 × ( 13 ) ( C N − N ) + 3 N + 3 N ( 1 N ) ]= A. (3.37)Case 4: q N +1 , q collide at t = 0.By (3 . q N +1 , q collide at t = ir , i = 0 , . . . , r − . (3 . . 5, (3 . Z ( 12 | ˙ q ( t ) − ˙ q N +1 ( t ) | + N + 3 | q ( t ) − q N +1 ( t ) | ) dt = r − X i =0 Z t i +1 t i ( 12 | ˙ q ( t ) − ˙ q N +1 ( t ) | + N + 3 | q ( t ) − q N +1 ( t ) | ) dt ≥ × (4 π )( N + 3) r ( 1 r ) . (3.39)From (3 . , (3 . − (3 . q N +2 , q , collide at t = ir + 23 ( mod q N +3 , q collide at t = ir + 13 ( mod , i =0 , . . . , r − ,q N +1 , q collide at ir + N − N ( mod q N +2 , q collide at ir + N − N + 23 ( mod q N +3 , q collide at ir + N − N + 13 ( mod , i = 0 , . . . , r − , ... q N +1 , q N − collide at ir + 2 N ( mod q N +2 , q N − collide at ir + 2 N + 23 ( mod q N +3 , q N − collide at ir + 2 N + 13 ( mod , i = 0 , . . . , r − ,q N +1 , q N collide at ir + 1 N ( mod q N +2 , q N collide at ir + 1 N + 23 ( mod q N +3 , q N collide at ir + 1 N + 13 ( mod , i = 0 , . . . , r − . Then by Lemma 2 . 5, Lemma 3 . 3, Remark 3 . 1, we have ∀ ≤ i ≤ r − , ≤ j ≤ , Z ( 12 | ˙ q i ( t ) − ˙ q N + j ( t ) | + N + 3 | q i ( t ) − q N + j ( t ) | ) dt ≥ × (4 π )( N + 3) r ( 1 r ) . (3.40)13o we get f ( q ) = 1 N + 3 X ≤ i 1, and (3 . q N +1 , q N +2 collide at t i = iN r , i = 0 , . . . , N r − . (3 . Z ( 12 | ˙ q N +1 ( t ) − ˙ q N +2 ( t ) | + N + 3 | q N +1 ( t ) − q N +2 ( t ) | ) dt = Nr − X i =0 Z t i +1 t i ( 12 | ˙ q N +1 ( t ) − ˙ q N +2 ( t ) | + N + 3 | q N +1 ( t ) − q N +2 ( t ) | ) dt ≥ × (4 π )( N + 3) N r ( 1 N r ) . (3.44)By(3 . q N +2 , q N +3 , collide at t = iN r + 23 , i = 0 , . . . , N r − , (3.45)14 N +3 , q N +1 collide at t = iN r + 13 , i = 0 , . . . , N r − . (3.46)Then by Lemma 2 . 5, Remark 3 . 1, (3 . . Z ( 12 | ˙ q N +2 ( t ) − ˙ q N +3 ( t ) | + N + 3 | q N +2 ( t ) − q N +3 ( t ) | ) dt ≥ × (4 π )( N + 3) N r ( 1 N r ) (3.47) Z ( 12 | ˙ q N +3 ( t ) − ˙ q N +1 ( t ) | + N + 3 | q N +3 ( t ) − q N +1 ( t ) | ) dt ≥ × (4 π )( N + 3) N r ( 1 N r ) . (3.48)So, we obtain f ( q ) = 1 N + 3 X ≤ i N r ) + 3 × ( 13 ) C N + 3 N ] △ = D. (3.49)When N is odd, let ˜ A = inf { A, C, D } , then on the collision set, the action functional f ≥ ˜ A .When N is even, let ˜ B = inf { A, B, C, D } , then on the collision set, the action func-tional f ≥ ˜ B .(1)Take N = 4 , d = 3 , r = 7 , k = 3 , k = − a > , b > 0, and q i = a (cos (6 πt + 2 π ( i − , sin (6 πt + 2 π ( i − , i = 1 , . . . , ,q j = b (cos ( − πt + 2 π ( j − , sin ( − πt + 2 π ( j − , j = 5 , , . 15e choose a = 0 . , b = 0 . A ≈ . , B ≈ . , C ≈ . , D ≈ . , ˜ B = 138 . ,f ( q ) ≈ . < ˜ B. This proves that the minimizer of f ( q ) on the closure ¯Λ is a non-collision solution of theseven-body problem.(2)Take N = 5 , d = 3 , r = 8 , k = 3 , k = − a > , b > 0, and q i = a (cos (6 πt + 2 π ( i − , sin (6 πt + 2 π ( i − , i = 1 , . . . , ,q j = b (cos ( − πt + 2 π ( j − , sin ( − πt + 2 π ( j − , j = 6 , , . We choose a = 0 . , b = 0 . A ≈ . , C ≈ . , D ≈ . , ˜ A = 181 . ,f ( q ) ≈ . < ˜ A. This proves that the minimizer of f ( q ) on the closure ¯Λ is a non-collision solution ofthe eight-body problem.(3)Take N = 7 , d = 3 , r = 10 , k = 3 , k = − a > , b >