Calculous Of Variations Algorithm For Periodic Solutions Of N-Body Problems

In this paper, we mainly consider the calculous of variations algorithm for periodic solutions of N-body problem.The N-body problem can be described with second order differential equations in accordance with Newton's law:where m_i is the mass of i-th body, q_i : R → R~d is the position of ?'-th body in Euclid space, N is the number of bodies, and G is the universal gravitational constant, which we always take to be 1 for convenience henceforth. LetwhereThen the problem of the periodic solutions for N-body can be transformed into the following boundary value problems with singular potential:(P)where q = (q\, ? ? ? ,qn) G (Rd)N, T is the period and the singular potential can be defined as follows.Definition 1 We call U a singular potential ifU? e C2(Rd\{0}, R), Ua(0 = Ujt(t), (Al)Uu(Z)->-oo, as |(hO, (A2),qN) G (i?d)N. (.43)In order to deal with the obstruction coming from the singularity of Newton potential, Gordon introduced the strong force condition.Definition 2 Uu is said to satisfy Strong Force Condition if there exist Vu € C1 (Rd\{0}, R), such that Vu —> +qo (￡ —>? 0)and(SF) w/iere |￡| isWe disperse the interval [0, T] into k parts with the step h, then we have the difference equations of N-body problems (N)-(P). LetS = {tj\tj=jh, j = 0, ???,*}, (2)where k is the number of the dispersed parts and the step length is h — T/k. Let ts+i, ts, ts_i G 5, the forward difference scheme can be defined with= q{ta+i) -q(ts),Then the derivative of q for ts can be approached withq{ts) = ^Aq(ts), q{ts) = ^A2q{ts). For the curve q, we introduce the linear difference schemet-L^q{s-i) + t-^=^q(s), *e[t._i,t.]. (DS)where q(s) is the abbreviation of q(ts), ts & S and q = (qi,--- , gjv)- Then the boundary value difference equations of (N) is the following\m-q- V4i{s)Uh = 0, (Nl)?(0) = $(*), 9(1) = g(* + 1), (PI)where M is a corresponding positive matrix, andk k(j = YJU{q{s)) = -Y, ￡ ^(?(?)-ft(*)). (3)is singular. Then the Lagrangian of (Nl)-(Pl) isj (4)Let /c = 2A;in (2), k € -^+. We introduce the radial symmetry Ao, which is defined bellow:Sl = {q\qe(Rd)N, s.t. qi^qh Vz # /},GA, 9(s + fc) = -g(s)}. (5)Theorem 1 Suppose Uu and U in (3) satisfy the assumptions (A\)-(AZ), and q satisfies the radial symmetry condition. Then (4) has infinitely many critical points, and these critical points correspond to general solutions of system (Nl)-(Pl);moreover if U satisfies (SF), then these critical points correspond to non-collision solutions of (Nl)-(Pl).We consider a class of multi-radial symmetry solutions bellow. Let Rd = Rd" x R^'. We set k = 4fc = Ik, k, k € Z+, for the difference system (Nl)-(Pl), and defineThen we have the following proposition, which is similar to Theorem 1.Proposition 1 Suppose q satisfies multi-radial symmetry A\, then system (Nl)-(Pl) has infinitely many general solutions;moreover if U satidfies Strong Force condition (SF), then (Nl)-(Pl) has infinitely many non-collision solutions.We also consider the Eight-shape solution from A. Chenciner and R. Montg-mery. Introduce the action Z2 x Z2 on R/TZ and R2:where t 6 i?, (^^,9^) is the coordinates of q in plane R2.Let g : {R/TZ) -> /?2, i = 1,2,3, and define the orbit of ith body byft = ?(* + (3-i)|)- (HO)SupposeT ----) + Q(t ~ 32T 6(HI)q(a(t))q(r(t))=r(q(t)),(H2)(i?2)3let l{t) :^(*) = (9i(*),92(*),93(*)), *G[0,T]. (H3)Then we haveProposition 2 (Nl)-(Pl) has infinitely many collision free solutions if q satisfies (H0)-(H3).For the case of iV > 3, we suppose (H2) is satisfied, andi = l,.-.,JV, (H4)N>(*) = °' (H5)J, I ￡ [U, 1 J l(t) . It 11 Zj —t \ix ) . l^l'-'JIf TV is even, then we can derive the following proposition:Proposition 3 Suppose (H2),(H4)-(H6), and N e Z+ is even. Then the i-th (i < y) body and (i + ^)-th body collide on original point at the same time t = jjT.If TV is an odd, we make a further assumption to the Eight-Shape orbit:q{t1)=q{t2) =*? tiGJo, ||, t = l,2. (H7)Proposition 4 Suppose (H2),(H4)-(H7), and N > 3 is odd. Then (Nl)-(Pl) has infinitely many collision free solutions.In order to find the saddle points of J, we search two sequences, which approach to the saddle points. By introducing these definitions of gradient flow and level, we prove the following theorem:Theorem 2 Let Jc~ be a disconnected sublevel of the functinalJ, Ff~ be the disjoint connect components of Jc~, and J-^~ be their basions of attraction. Let qi ￡ F±~, i = 1,2. Then for all the points on the ridge of Ff~, there exist two sequences start from Qi and qi, such thatlimBasing on these two sequences, we prove the following Mountain Pass theorem:Theorem 3 Let J be a C2 functional on a Hilbert space H. Let 91,92 € H, Tquq2 be the set of paths and Cq be the levelc0 = inf 7 G rgii92 sup J(t(s)), (6)?6[0,l]such thatco>max{J(q1)1J{q2)}. (7)If the functional of J satisfies the P-S condition at level cq, then there exists a critical point for the functional J at level Cq, and it is not a local minimizer.At the last part, we describe the algorithms and present some examples.