Research Article The Characterization of the Variational Minimizers for Spatial Restricted N+1-Body Problems

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1 Abstract and Applied Analysis Volume 23, Article ID , 5 pages Research Article he Characterization of the Variational Minimizers for Spatial Restricted +-Body Problems Fengying Li, Shiqing Zhang, and Xiaoxiao Zhao Yangtze Center of Mathematics and College of Mathematics, Sichuan University, Chengdu 664, China Correspondence should be addressed to Fengying Li; lify38@63.com Received 8 February 23; Accepted 27 April 23 Academic Editor: Maoan Han Copyright 23 Fengying Li et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We use Jacobi s necessary condition for the variational minimizer to study the periodic solution for spatial restricted +-body problems with a zero mass on the vertical axis of the plane for equal masses. We prove that the minimizer of the Lagrangian action on the anti-/2 or odd symmetric loop space must be a nonconstant periodic solution for any 2 472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. his result contradicts with our intuition that the small mass should always be at the origin.. Introduction and Main Result he ewtonian n-body problem is a classical problem. Spatial restricted 3-body model was studied by Sitnikov 2. Mathlouthi 3 et al. studied the periodic solutions for the spatial circular restricted 3-body problems by mini-max variational methods. In this paper, we study spatial circular restricted +- body problems with a zero mass on the vertical axis of the plane for equal masses. Suppose point masses m = = m =move on a circular orbit around the center of masses. he motion for the zero mass is governed by the gravitational forces of m,...,m.letρ j =e (2πj/) and q (t) =re 2πt ρ,...,q j (t) () =ρ j q (t),..., q (t) =re 2πt satisfy the ewtonian equations: m i q i = U, i=,...,, (2) q i U= i<j m i m j q. (3) i q j he orbit q(t) = (,, z(t)) R 3 for zero mass satisfies the following equation: q= i= m i (q i q) q i q 3. (4) Obviously, q(t) = (,, ) satisfies (4); it seems that q(t) is a variational minimizer, but we will prove it is not this is the goal of this paper. Define then f(q)= 2 q 2 + i= q q dt, q Λ i, (5) i f (q) = 2 2 z + r 2 +z dt f (z), q Λ i, (6) 2

2 2 Abstract and Applied Analysis Λ = q (t) = (,, z (t)) z(t) W,2 ( R Z,R) } }, q ( t) = q(t) } Λ 2 = q (t) = (,, z (t)) z(t) W,2 ( R Z,R) } z(t+, 2 )= z(t) } } W,2 ( R Z,R)=x(t) x (t), x (t) L 2 ((, ),R) }. x (t+) =x(t) otice that the symmetry in Λ 2 is related to the Italian symmetry 4. In this paper, our main result is the following. heorem. he minimizer of f(q) on the closure Λ i of Λ i (i =, 2) is a nonconstant periodic solution for 2 472; hence thezeromassmustoscillate,sothatitcannotbealwaysinthe same plane with the other bodies. 2. Proof of heorem We define the inner product and equivalent norm of W,2 (R/Z, R): u, V = (uv +u V )dt, /2 u = u 2 dt which is equivalent to /2 + u 2 dt, /2 u 2 dt + u (). (9) Lemma 2 (Palais Symmetry Principle 5). By Palais Symmetry Principle, we know that the critical point of f(q) in Λ i is a noncollission periodic solution of ewtonian equation (4). Let σ be an orthogonal representation of a finite or compact group G in the real Hilbert space H such that for all σ G,f(σ x) = f(x),f:h R. Let S=x H σ x=x,for all σ G}. hen the critical point of f in S is also a critical point of f in H. In order to prove heorem, we need the following lemmas: Lemma 3 (see 6). Let X be a reflexive Banach space, S be a weakly closed subset of X, f : S R + },andf + is weakly lower semicontinuous and coercive (f(x) + as x + ); then f attains its infimum on S. Lemma 4 (Poincare-Wirtinger Inequality). Let q W,2 (R/Z, R ) and q(t)dt = ;then 2 dt ( 2π 2 ) q(t) 2 dt. () (7) (8) Lemma 5. f(q) in (6) attains its infimum on Λ =Λ or Λ 2 = Λ 2. Proof. By Lemmas 3 and 4,itiseasytoproveLemma 5. Lemma 6 (Jacobi s ecessary Condition 7). If the critical point u = u(t) corresponds to a minimum of the functional b F(t, u(t), a u (t))dt and if F u u >along this critical point, then the open interval (a, b) contains no points conjugate to a; that is, for all c (a, b), the following boundary value problem d dt (Ph )+Qh=, h (a) =, h(c) =, has only the trivial solution h(t),forallt (a,c), P= 2 F u u u= u Q= 2 (F uu d dt F uu ) u= u. () (2) Remark 7. It is easy to see that Lemma 6 is suitable for the fixed end problem. In this paper, we consider the periodic solutions of (2) onλ i = Λ i (i =, 2); hence we need to establish a similar conclusion as Lemma 6 for the periodic boundary problem. Lemma 8. Let F C 3 (R R R,R). Assume that u= u(t) is a critical point of the functional F(t, u(t), u (t))dt on u W,2 (R/Z, R), u () = } and F u u u= u >.Iftheopen interval (, ) contains a point c conjugate to,thenu= u(t) is not a minimum of the functional F(t, u(t), u (t))dt. Proof. Suppose u = u(t) is a minimum of the functional F(t, u(t), u (t))dt. he second variation of F(t, u(t), u (t))dt is (Ph 2 +Qh 2 )dt, (3) P= 2 F u u u= u, Q= 2 (F uu d dt F uu ) u= u. (4) Set Q u (h) = (Ph 2 +Qh 2 )dt. (5) For all h u W,2 (R/Z, R), u () = }, itiseasyto see that Q u (h). henbyq u (θ) =, θ is a minimum of Q u (h).heeuler-lagrangeequationwhichiscalledthejacobi equation of (5)is d dt (Ph ) +Qh=. (6)

3 Abstract and Applied Analysis 3 Since the interval (, ) contains a point c conjugate to,there exists a nonzero Jacobi field h C 2 (,, R) satisfying Letting d dt (Ph )+Qh =, (7) h () =, h (c) =, h () =. hen herefore r 3 = 6π 2 r=( 2/3 4π ) csc ( π j). (24) j csc ( π j). (25) j /3 ĥ (t) = h (t) t, c, t (c,, we have ĥ C2 (, \ c}, R), ĥ() = ĥ(c) = ĥ() = and Q u (ĥ) = (Pĥ 2 +Qĥ2 )dt (8) = (Ph 2 +Qh2 )dt=. (9) c otice that we can extend ĥ periodically when we take as the period, so ĥ W,2 (R/Z, R). Forallh C (,, R), it is easy to check that Q u (h). henby(9), one has ĥ C 2 (, \ c}, R) W,2 (R/Z, R) is a minimum of Q u (h). Henceweget d (Pĥ )+Qĥ=. (2) dt Combining with ĥ() = ĥ(c) = and ĥ () =, by the uniqueness of initial value problems for secondorder differential equation, we have ĥ(t) on, c, which contradicts the definition of ĥ. herefore, Lemma 8 holds. Lemma 9. he radius r for the moving orbit of equal masses is r=( 2/3 4π ) Proof. By () (3), we have j Substituting ()into(22), we have csc ( π j) /3. (2) q j q q = j = q 3. (22) j q 4π 2 = j = 4π 2 r 3 ρ j = j = ρ 3 = 4 j ρ j ρ r 3 ρ j ρ j 3, csc ( π j). (23) Proof of heorem. Clearly, q(t) = (,, ) is a critical point of f(q) on Λ i =Λ i (i =, 2).Forthefunctional(6), let F (z, z ) = 2 2 z + r 2 +z. 2 (26) hen the second variation of (6) in the neighborhood of z= is given by (Ph 2 +Qh 2 )dt, (27) P= 2 F z z = z= 2, Q= 2 (F zz d dt F zz ) = (28) z= 2r 3. he Euler equation of (27) is called the Jacobi equation of the original functional (6), which is that is, d dt (Ph )+Qh=, (29) h + h=. (3) r3 ext, we study the solution of (3)withinitialvaluesh() =, h () =.Itiseasytoget h (t) = r3 sin r 3 t, (3) which is not identically zero on,,butwewillproveh(c) = for some c (, ). Suppose there exists c (, ) such that h(c) =. Hence, for some k Z + We have by using (24). c= π2 k 2 r 3 /2 k 2 ( j= = r 3 c=kπ. (32) /2 (33) csc ((π/) j)) 6

4 4 Abstract and Applied Analysis Case (Minimizing f(q) on Λ =Λ ). Letting <c</2, h (t) t, c, h (t) = t (c, c, h ( t) t ( c,. (34) It is easy to check that h(t) C 2 (, \ c, c}, R) W,2 (R, R), h( t) = h(t), h() = h() =, h(c) = h(c) =, and h is a nonzero solution of (3). If we take k=, c= It is equivalent to Let j= j= csc ((π/) j) 6 /2 < 2. (35) csc ( π j) < 4. (36) f (x) = csc ( π x j). (37) j =x It is not hard to check that f(x) is nonmonotone. But for 2 472,(36) holds by writing program to calculate it. herefore, for 2 472,wehavec (, ) such that sin r 3 c=sin π=. (38) otice that we can extend h periodically when we take as the period, so h Λ.henbyLemma 8, q(t) = (,, ) is not a local minimum for f(q) on Λ. Hence the minimizers of f(q) on Λ arenotalwaysatthecenterofmasses;they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; therefore, we get the nonplanar periodic solutions. Case 2 (Minimizing f(q) on Λ 2 =Λ 2 ). Let h (t) t, c, t (c, 2, h (t) = h (t 2 ) t ( 2, 2 +c, t ( 2 +c,. (39) It is easy to check that h(t) C 2 (, \c, /2, (/2)+c}, R) W,2 (R, R), h(t + (/2)) = h(t), h() = h() =, h(c) = h(c) =,and h is a nonzero solution of (3). We hope c (, /2);thatis, c= j= csc ((π/) j) 6 /2 < 2. (4) It implies that j= csc ( π j) < 4. (4) Calculated by program, for 2 = 472, wehave c (, /2) such that h(c) =. otice that we can extend h periodically when we take as the period, so h Λ 2.henbyLemma 8, q(t) = (,, ) is not a local minimum for f(q) on Λ 2. Hence the minimizers of f(q) on Λ 2 are not always at the center of masses; they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; therefore, we get the nonplanar periodic solutions. We can use another argument to get much larger.we construct a test function z(t) such that f(z) < f() = /r for, is a very large number. Let t t, 4, z (t) = t 4 4, 2, (42) and we extend z(t) by z(t + (/2)) = z(t).wehave f (z) = / r 2 +z dt 2 = 4 +2 /4 /2 r 2 +t dt /4 r 2 + (/6) dt = 4 +2ln + + 6r 2 4r r. 2 (43) Writing program to calculate, we find an =9 such that f(z) < f(). Hence q(t) = (,, ) is not a local minimum for f(q) on Λ i =Λ i (i =, 2). So the minimizers of f(q) on Λ i are not always at the center of masses; they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; hence, we get the nonplanar periodic solutions. Acknowledgments he authors would like to thank the referee for his/her many valuable comments and suggestions. his paper is Supported by the national atural Science Foundation of China (775) and the Ph.D Programs Foundation of Ministry of Education of China. References A. Wintner, he Analytical Foundations of Celestial Mechanics, vol. 5 of Princeton Mathematical Series, Princeton University Press, Princeton, J, USA, K. Sitnikov, Existence of oscillating motions for the three-body problem, Doklady Akademii auk USSR,vol.33,pp , 96.

5 Abstract and Applied Analysis 5 3 S. Mathlouthi, Periodic orbits of the restricted three-body problem, ransactions of the American Mathematical Society, vol. 35, no. 6, pp , U. Bessi and V. Coti Zelati, Symmetries and noncollision closed orbits for planar -body-type problems, onlinear Analysis. heory, Methods & Applications, vol. 6, no. 6, pp , R. S. Palais, he principle of symmetric criticality, Communications in Mathematical Physics,vol.69,no.,pp.9 3, M. Struwe, Variational Methods,vol.34ofResults in Mathematics and Related Areas, Springer, Berlin, Germany, 3rd edition, 2. 7 I. Gelfand and S. Formin, Calculus of Variations, auka, Moscow, Russia; Prentice-Hall, Englewood Cliffs, J, USA, English edition, 965.

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