- Email: [email protected]
A Noncollision Periodic Solution for N-Body Problems
Journal of Mathematical Analysis and Applications 257, 332–342 (2001) doi:10.1006/jmaa.2000.7349, available online at http://www.idealibrary.com on
A Noncollision Periodic Solution for N-Body Problems1 Shiqing Zhang Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China; and Department of Mathematics, Chongqing University, Chongqing 400044, People’s Republic of China
and Qing Zhou Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China Submitted by F. E. Udwadia Received April 11, 2000 Using variational minimization methods, we prove the existence of one noncollision periodic solution for N-body type problems whose potentials are pinched between two homogeneous potentials in R k (k ≥ 2). © 2001 Academic Press Key Words: N-body problems; noncollision periodic solution; variational methods.
1. INTRODUCTION AND MAIN RESULTS The motion of N-body type problems [1, 2, 9, 12, 19, 20] is related with solving the following second order differential equations, ∂U mi q¨ i = (1.1) ∂qi where mi > 0 is the mass of the ith body and qi ∈ R k k ≥ 2 is the position of the ith body, and the potential Uq = Uq1 qN = Uij qi − qj (1.2) 1≤i
where Uij x ∈ C 1 R k \0 R. 1
Partially supported by grants of MOST, NSFC, and QSSTF. 332
0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
noncollision periodic solutions
333
In the last 20 years, some researchers applied variational methods to study the periodic solutions of N-body type problems [3–8, 10, 13, 14, 18, 21, 22], but they didn’t get the existence of one noncollision periodic solution for any given masses of N bodies. Observing the symmetry and choosing a suitable domain of the Lagrangian action integral for (1.1), we prove that the minimizer of the Lagrangian action integral is one noncollision periodic solution of (1.1)–(1.2) assuming the potential Uq is pinched between two homogeneous potentials. Let Okk ≥ 2 denote the rotational group in R k and Bθ 0 ∈ Ok Aθ = (1.3) 0 −Ik−2 θ ∈ 0 2π where Ik−2 is a unit matrix with order k − 2 and cos θ − sin θ Bθ = sin θ cos θ Let
(1.4)
(1.5) H = W 12 R/TZ R k 2π T H# = x ∈ H x t + =A xt r ≥ 2 an integer (1.6) r r E = q = q1 qN qi − qj ∈ H# i j = 1 N N
= q = q1 qN ∈ E mi qi t ≡ 0 E (1.7)
i=1
i t = qj t = q = q1 qN ∈ E q
∀ t ∈ R 1 ≤ i = j ≤ N f q =
(1.8)
T N 1 T mi q˙i 2 dt + Uqdt 2 0 i=1 0
(1.9)
Theorem 1.1. Assume Uq satisfies (1) mi mj mi mj b a ≤ Uq ≤ α 2 1≤i
2 1≤i
α>0
(1.10)
334
zhang and zhou (2) UAq = Uq
(1.11)
for some A = A2π/r ∈ Ok where r will be defined later. Then there is an integer r depending on α and masses m1 mN such that the minimizer of f q on is one noncollision T-periodic solution for (1.1)–(1.2). Theorem 1.2. If α = 1 N = 3, m1 = m2 = m3 = 1, and a = b = 1 then the minimizer of f q on with r = 2 is one noncollision T-periodic solution. Remark. The domain for f q is different from the one defined by Bessi and Coti Zelati [4]. 2. THE PROOF OF THEOREM 1.1 In order to prove Theorem 1.1, first we give some lemmas: Lemma 2.1. The critical points of f q in are noncollision T-periodic solutions of (1.1)–(1.2).
for f q restricted on E
is First, we prove that the critical point q ∈ E also a critical point for f q on E. In fact, the condition that the center of masses is fixed at the origin is equivalent to N 2 gq = mi qi t ≡ 0 (2.1) i=1
Hence by the Lagrangian multiplier rule, for any critical point q of f q on
we have E, f q + λg q = 0
(2.2)
that is, for any ϕ = ϕ1 ϕN , ϕi ∈ W 12 R/TZ R k we have f q ϕ + λg q ϕ = 0 N N f q ϕ + 2λ mi qi mi q˙i ϕi = 0 i=1
i=1
(2.3) (2.4)
noncollision periodic solutions
335
f q ϕ = 0
(2.5)
that is,
Now assume q ∈ E is a critical point of f q on E. Then f q y = 0
∀ y ∈ E
(2.6)
Hence we have p = p1 pN ∈ E ⊥ , where pi ≡ q¨ i −
∂U ∂qi
i = 1 N
(2.7)
On the other hand, we notice that pi − pj = q¨ i − q¨ j −
∂U ∂U + ∂qi ∂qj
= qi − qj − +
N aαmi mj qj − qi j =ij=1
qi − qj α+2
N aαmi mj qi − qj i =ji=1
qj − qi α+2
(2.8)
Hence pi − pj ∈ H# p ∈ E. Hence p ∈ E ⊥ ∩ E = 0; that is, q is a solution of (1.1). Lemma 2.2. The functional f is coercive on ; that is, for any qn qn H → +∞ f qn → +∞. Proof.
For any q = q1 qN ∈ we have T T T ≡ qi t + − qj t + qi − qj t + r r r 2π =A qi t − qj t r 2π ≡A qi − qj t r 2 qi − qj t + T − qi − qj t t 2 2π = A qi − qj t − qi − qj t r 2 π 2 = 2 sin qi − qj t r
(2.9)
(2.10)
336
zhang and zhou
On the other hand, 2 qi − qj t + T − qi − qj t r T 2 t+ r = q˙i − q˙j dt t
T ≤ r =
t+ Tr
t
2
q˙i − q˙j t dt
2 T T q˙i − q˙j t dt 2 r 0
(2.11)
Hence we have 2 T 2 q˙i − q˙j t 2 dt ≥ r 2 sin π qi − qj t 2 T r 0 T 2 mi mj q˙i − q˙j dt
(2.12)
0 1≤i
≥ M
π 2 r 2 m m q − qj 2 2 sin T r 1≤i
N 2 π 2 r 2 mi q˙i dt ≥ 2 sin M mi qi 2 T r i=1 i=1
N T
0
(2.13) (2.14)
where M=
N i=1
mi
(2.15)
Hence the standard norm for H is equivalent to q ˙ 2=
N 1
0 i=1
mi q˙i 2 dt
1/2
(2.16)
Hence the definition of f q implies f is coercive. Lemma 2.3. The system (1.1)–(1.2) has a weak T-periodic solution q = q1 qN ∈ in the sense of Bari and Rabinowitz [3]: (1◦ )
qi ∈ W 12 R/TZ R k .
(2◦ ) The collision set C = t ∈ 0 T qi t = qj t for some 1 ≤ i = j ≤ N has Lebesgue measure 0.
noncollision periodic solutions (3◦ )
337
qi is C 2 on 0 T \C and satisfies (1.1) and energy conservation, N 1 m q˙ 2 − Uq1 qN = h 2 i=1 i i
(2.17)
Proof. It’s easy to prove f q has positive lower bound and is weakly lower semi-continuous, so Lemma 2.2 implies Lemma 2.3. In order to get a good lower bound estimate of f q on the collision solutions, we need another lemma:
Lemma 2.4. Let index sets A and B satisfy A B = and A B = 1 2 N. Then
m i mj
qi − qj α ij∈A×B
≥
ij∈A×B
mi m j
1+ α2
ij∈A×B
mi mj qi − qj
2
− α2
(2.18)
For the proof of Lemma 2.4 refer to Long and Zhang [13]. In order to facilitate the reader, we repeat it. By H¨ older’s inequality, we have 2 mi mj α (2.19) mi mj ≤ m m
q − q
i j i j
qi − qj α i∈A j∈B i∈A j∈B i∈A j∈B By H¨ older’s inequality, we have mi mj qi − qj α i∈A j∈B
≤
i∈A j∈B
mi m j
2−α 2
i∈A j∈B
mi mj qi − qj 2
α/2
(2.20)
By (2.19) and (2.20) we get (2.18). Now we estimate the lower bound of f q on the collision solutions. Let SN denote the group of all the permutations of 1 N. For l = 2 N, we set ∂ l = q ∈ E ∃ s ∈ SN ∃ t¯ ∈ 0 T s.t. qs1 t¯ = · · · = qsl t¯ (2.21) First, we assume l = 2, s is the identity, and t¯ = 0. Then by the Lagrangian identity and the symmetry property qi − qj t + Tr = A 2π qi − qj t r we have f q ≥ g1 q + g2 q + g3 q
(2.22)
338
zhang and zhou
where
g1 q = r
× g2 q = r
g3 q = r
1≤i =j≤2 T/r
1 1 2 dt
q˙ − q˙j + Ma 2 i
qi − qj α
3≤i =j≤N T/r
0
m i mj
1 1 dt
q˙i − q˙j 2 + Ma 2
qi − qj α
0
(2.24)
mi m j
1≤i≤2 3≤j≤N T/r
(2.23)
mi mj
2 2M ×
0
1 2M ×
1 2M
1 1 dt
q˙i − q˙j 2 + Ma 2
qi − qj α
(2.25)
Using the estimates of the Lagrangian action integral on collision solutions of two body [7] problems we have Lemma 2.5. 2α
2−α
g1 q ≥ C1 r 2+α T 2+α where −α
C1 = AM22+α M2 =
2 i=1
1≤i =j≤2
mi m j
mi
(2.26)
(2.27) (2.28)
2α 1 1 1 A= + αa2/α+2 2π 2+α 2 2 α Let
1≤i =j≤2
B2 = mins∈SN
2 i=1
(2.29)
msi msj
msi
α/2+α
= AB C 1 2
(2.30)
(2.31)
Then 2α 2−α r 2+α T 2+α infg1 q q ∈ ∂ 2 ≥ C 1
(2.32)
noncollision periodic solutions
339
By the arguments of Degiovanni and Giannoni [7], we can get the lower bound estimate on g2 q, 2−α
g2 q ≥ C2 T 2+α
(2.33)
where −α
2+α C2 = AMN−2
MN−2 = Let
N i=3
N i=3
mi mj
(2.34)
mi
(2.35)
3≤i =j≤N msi msj BN−2 = min N α/2+α s∈SN i=3 msi
(2.36)
= AB C 2 N−2
(2.37)
Then T 2+α infg2 q q ∈ ∂ 2 ≥ C 2 2−α
(2.38)
We use inequality (2.18) of Lemma 2.4, Sundman’s inequality [19], and the arguments of Degiovanni and Giannoni [7] to estimate the lower bound for g3 q: 1+ α2 1 T 2 g3 q ≥ m m q˙ − q˙j dt + a m i mj 2M 0 1≤i≤2 3≤j≤N i j i 1≤i≤2 3≤j≤N
×
1≤i≤2 3≤j≤N
≥
1 T 2M 0 +a
mi mj qi − qj 2
m i mj
1≤i≤2 3≤j≤N
×
T 0
dt
(2.39)
1/2 2 d 2 dt m m
q − q
i j i j dt 1≤i≤2 3≤j≤N
−α/2
1≤i≤2 3≤j≤N
1+ α2
mi mj qi − qj 2
−α/2
dt
1+ α2 1 T 2 ≥ inf
r
˙ dt + a mi m j 2M 0 1≤i≤2 3≤j≤N
(2.40)
×
T
0
r −α dt r ∈ W 12 0 T R+
(2.41)
340
zhang and zhou
1 = T inf 2M
2π T
2
R +a 2
1≤i≤2 3≤j≤N
×
mi mj
1+ α2
1 R > 0 Rα
(2.42)
2 2 α+2 1+ α2 α+2 α α 2 1 2π 2 α+2 = 1+ a m i mj 2 α 2M T 1≤i≤2 3≤j≤N 2 α+2 α α 2 2−α 2 2 α+2 α − α+2 1+ = M a α+2 mi mj T 2+α 2π 2 α 1≤i≤2 3≤j≤N 2 2α α 2−α 2 1 1+ αa α+2 2π α+2 M − α+2 = mi mj T 2+α 2 α 1≤i≤2 3≤j≤N α 2−α 2−α = 2AM − α+2 mi mj T 2+α ≡ C3 T 2+α (2.43)
1≤i≤2 3≤j≤N
Let B=
mins∈SN
1≤i≤2 3≤j≤N M α/α+2
msi msj
(2.44)
= 2AB C 3
(2.45)
Then T 2−α 2+α infg3 q q ∈ ∂ 2 ≥ C 3 r inff q q ∈ ∂ 2 ≥ C 1
2α 2+α
(2.46) +C T +C 2 3
2−α 2+α
(2.47)
r 2+α + C +C T 2+α inff q q ∈ ∂ l ≥ C 1 2 3
(2.48)
It’s easy to see that for l > 2 we also have 2α
2−α
Remark. The corresponding lower bound estimate in Bessi and Coti Zelati [4] is not correct since the symmetry breaks down when they move the binary collision to the origin and they work on their domain. Lemma 2.6 [6]. Let ρ=
m i mj
sinπi − j/N α 2 πi − j σ= mi mj sin N 1≤i =j≤N
(2.49)
1≤i =j≤N
(2.50)
noncollision periodic solutions
341
Then the minimizing value for Lagrangian action f q has upper bound estimate 2 2α 2 α 2−α 1 1 1 f q ≤ + bα 2+α 2π α+2 ρ α+2 σ α+2 M −α/α+2 T 2+α 2 2 α
2−α 2+α ≡ CT
(2.51)
Now we can prove Theorem 1.1. Assume the minimizer q for f q on has a collision time t¯ ∈ 0 T . + r 2α/2+α + C Then by Lemma 2.5, we can get a lower bound estimate C 1 2 2−α/2+α T for f q, which depends on the integer r ≥ 2. By Lemma 2.6 C 3 we can choose r large enough so that 2α +C > C
r 2+α +C C 1 2 3
(2.52)
This is a contradication. Now we prove Theorem 1.2. If α = 1 m1 = m2 = m3 = 1 a = 1, and r = 2 then we have A=
3 2π2/3 4
B2 = 22/3 = 3 · 2−4/3 2π2/3 C 1
2α r 2+α C = 3 · 2−2/3 2π2/3 1
=0 C 2 B = 2 · 3−1/3 2α r 2+α C 1
= 2AB = 32/3 2π2/3 C 3 + C2 + C3 = 3 · 2−2/3 + 32/3 2π2/3
On the other hand, we compute
1 2 = √ 6 = 4 · 31/2
sinπi − j/3
3 1≤i =j≤3 π 2 9 πi − j 2 σ= = 6 sin = sin 3 3 2 1≤i =j≤3 ρ=
1/3 13 2/3 1/2 2/3 9
C= 2π 4 · 3 · 3−1/3 22 2
1 5/3 3 2π2/3 < 3 · 2−2/3 + 32/3 2π2/3 2 2α r 2+α +C =C +C =
1
2
3
342
zhang and zhou REFERENCES
1. R. Abraham and J. Marsden, “Foundations of Mechanics,” 2nd ed., Benjamin/Cummings, London, 1978. 2. V. Arnold, V. Kozlov, and A. Neishtadt, “Dynamical Systems. III. Mathematical Aspects of Classical and Celestial Mechanics,” Russian ed., 1985; English ed., Springer-Verlag, Berlin, 1988. 3. A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of three-body type, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 8 (1991), 561–649. 4. U.Bessi and V. Coti Zelati, Symmetries and noncollision closed orbits for planar N-bodytype problems, Nonlinear Anal. 16 (1991), 587–598. 5. A. Chenciner and N. Desolneux, Minima de l’int´egrale d’action et ´equilibres relatifs de n corps, C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), 1209–1212. 6. V. Coti Zelati, The periodic solutions of N-body type problems, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 7 (1990), 477–492. 7. M. Degiovanni and F. Giannoni, Dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup. Pisa 15 (1989), 467–494. 8. G. F. Dell’Antonio, Classical solutions of a perturbed N-body system, in “Topological Nonlinear Analysis, II” (M. Matzeu and A. Vignoli, Eds.), pp. 1–86, Birkh¨auser, Basel, 1997. 9. L. Euler, De motu rectilineo trium corp¨ orum se mutuo attrahentium, Novi. Comm. Acad. Sci. Imp. Petropll. (1767), 145–151. 10. W. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), 961–971. 11. G. Hardy, J. Littlewood, and G. Polya, “Inequalities,” 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1952. 12. J. Lagrange, Essai sur le probl´eme des trois corps, 1772, Ouvres 3 (1783), 229–331. 13. Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem, Abstract, Chinese Sci. Bull. 44, No. 18 (1999), 1653–1654; Acta Math. Sinica, (English series), 16 (2000), 579–592. 14. Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations 160 (2000), 422–438. 15. K. Meyer and G. Hall, “Introduction to Hamiltonian Systems and the N-Body Problems,” Springer-Verlag, Berlin, 1992. 16. H.Pollard, “Celestial Mechanics,” Amer. Math. Soc., Providence, 1976. 17. P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157–184. 18. E. Serra and S. Terracini, Collisionless periodic solutions to some three-body problems, Arch. Rational Mech. Anal. 120 (1992), 305–325. 19. C. Siegel and J. Moser, “Lectures on Celestial Mechanics,” Springer-Verlag, Berlin, 1971. 20. A. Wintner, “Analytical Foundations of Celestial Mechanics,” Princeton Univ. Press, Princeton, NJ, 1941. 21. S. Zhang and Q. Zhou, Geometric characterization for the least Lagrangian action of N-body problems, Science in China 44 (2001), 15–20. 22. S. Zhang and Q. Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sinica., to appear.
A Noncollision Periodic Solution for N-Body Problems1 Shiqing Zhang Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China; and Department of Mathematics, Chongqing University, Chongqing 400044, People’s Republic of China
and Qing Zhou Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China Submitted by F. E. Udwadia Received April 11, 2000 Using variational minimization methods, we prove the existence of one noncollision periodic solution for N-body type problems whose potentials are pinched between two homogeneous potentials in R k (k ≥ 2). © 2001 Academic Press Key Words: N-body problems; noncollision periodic solution; variational methods.
1. INTRODUCTION AND MAIN RESULTS The motion of N-body type problems [1, 2, 9, 12, 19, 20] is related with solving the following second order differential equations, ∂U mi q¨ i = (1.1) ∂qi where mi > 0 is the mass of the ith body and qi ∈ R k k ≥ 2 is the position of the ith body, and the potential Uq = Uq1 qN = Uij qi − qj (1.2) 1≤i
where Uij x ∈ C 1 R k \0 R. 1
Partially supported by grants of MOST, NSFC, and QSSTF. 332
0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
noncollision periodic solutions
333
In the last 20 years, some researchers applied variational methods to study the periodic solutions of N-body type problems [3–8, 10, 13, 14, 18, 21, 22], but they didn’t get the existence of one noncollision periodic solution for any given masses of N bodies. Observing the symmetry and choosing a suitable domain of the Lagrangian action integral for (1.1), we prove that the minimizer of the Lagrangian action integral is one noncollision periodic solution of (1.1)–(1.2) assuming the potential Uq is pinched between two homogeneous potentials. Let Okk ≥ 2 denote the rotational group in R k and Bθ 0 ∈ Ok Aθ = (1.3) 0 −Ik−2 θ ∈ 0 2π where Ik−2 is a unit matrix with order k − 2 and cos θ − sin θ Bθ = sin θ cos θ Let
(1.4)
(1.5) H = W 12 R/TZ R k 2π T H# = x ∈ H x t + =A xt r ≥ 2 an integer (1.6) r r E = q = q1 qN qi − qj ∈ H# i j = 1 N N
= q = q1 qN ∈ E mi qi t ≡ 0 E (1.7)
i=1
i t = qj t = q = q1 qN ∈ E q
∀ t ∈ R 1 ≤ i = j ≤ N f q =
(1.8)
T N 1 T mi q˙i 2 dt + Uqdt 2 0 i=1 0
(1.9)
Theorem 1.1. Assume Uq satisfies (1) mi mj mi mj b a ≤ Uq ≤ α 2 1≤i
2 1≤i
α>0
(1.10)
334
zhang and zhou (2) UAq = Uq
(1.11)
for some A = A2π/r ∈ Ok where r will be defined later. Then there is an integer r depending on α and masses m1 mN such that the minimizer of f q on is one noncollision T-periodic solution for (1.1)–(1.2). Theorem 1.2. If α = 1 N = 3, m1 = m2 = m3 = 1, and a = b = 1 then the minimizer of f q on with r = 2 is one noncollision T-periodic solution. Remark. The domain for f q is different from the one defined by Bessi and Coti Zelati [4]. 2. THE PROOF OF THEOREM 1.1 In order to prove Theorem 1.1, first we give some lemmas: Lemma 2.1. The critical points of f q in are noncollision T-periodic solutions of (1.1)–(1.2).
for f q restricted on E
is First, we prove that the critical point q ∈ E also a critical point for f q on E. In fact, the condition that the center of masses is fixed at the origin is equivalent to N 2 gq = mi qi t ≡ 0 (2.1) i=1
Hence by the Lagrangian multiplier rule, for any critical point q of f q on
we have E, f q + λg q = 0
(2.2)
that is, for any ϕ = ϕ1 ϕN , ϕi ∈ W 12 R/TZ R k we have f q ϕ + λg q ϕ = 0 N N f q ϕ + 2λ mi qi mi q˙i ϕi = 0 i=1
i=1
(2.3) (2.4)
noncollision periodic solutions
335
f q ϕ = 0
(2.5)
that is,
Now assume q ∈ E is a critical point of f q on E. Then f q y = 0
∀ y ∈ E
(2.6)
Hence we have p = p1 pN ∈ E ⊥ , where pi ≡ q¨ i −
∂U ∂qi
i = 1 N
(2.7)
On the other hand, we notice that pi − pj = q¨ i − q¨ j −
∂U ∂U + ∂qi ∂qj
= qi − qj − +
N aαmi mj qj − qi j =ij=1
qi − qj α+2
N aαmi mj qi − qj i =ji=1
qj − qi α+2
(2.8)
Hence pi − pj ∈ H# p ∈ E. Hence p ∈ E ⊥ ∩ E = 0; that is, q is a solution of (1.1). Lemma 2.2. The functional f is coercive on ; that is, for any qn qn H → +∞ f qn → +∞. Proof.
For any q = q1 qN ∈ we have T T T ≡ qi t + − qj t + qi − qj t + r r r 2π =A qi t − qj t r 2π ≡A qi − qj t r 2 qi − qj t + T − qi − qj t t 2 2π = A qi − qj t − qi − qj t r 2 π 2 = 2 sin qi − qj t r
(2.9)
(2.10)
336
zhang and zhou
On the other hand, 2 qi − qj t + T − qi − qj t r T 2 t+ r = q˙i − q˙j dt t
T ≤ r =
t+ Tr
t
2
q˙i − q˙j t dt
2 T T q˙i − q˙j t dt 2 r 0
(2.11)
Hence we have 2 T 2 q˙i − q˙j t 2 dt ≥ r 2 sin π qi − qj t 2 T r 0 T 2 mi mj q˙i − q˙j dt
(2.12)
0 1≤i
≥ M
π 2 r 2 m m q − qj 2 2 sin T r 1≤i
N 2 π 2 r 2 mi q˙i dt ≥ 2 sin M mi qi 2 T r i=1 i=1
N T
0
(2.13) (2.14)
where M=
N i=1
mi
(2.15)
Hence the standard norm for H is equivalent to q ˙ 2=
N 1
0 i=1
mi q˙i 2 dt
1/2
(2.16)
Hence the definition of f q implies f is coercive. Lemma 2.3. The system (1.1)–(1.2) has a weak T-periodic solution q = q1 qN ∈ in the sense of Bari and Rabinowitz [3]: (1◦ )
qi ∈ W 12 R/TZ R k .
(2◦ ) The collision set C = t ∈ 0 T qi t = qj t for some 1 ≤ i = j ≤ N has Lebesgue measure 0.
noncollision periodic solutions (3◦ )
337
qi is C 2 on 0 T \C and satisfies (1.1) and energy conservation, N 1 m q˙ 2 − Uq1 qN = h 2 i=1 i i
(2.17)
Proof. It’s easy to prove f q has positive lower bound and is weakly lower semi-continuous, so Lemma 2.2 implies Lemma 2.3. In order to get a good lower bound estimate of f q on the collision solutions, we need another lemma:
Lemma 2.4. Let index sets A and B satisfy A B = and A B = 1 2 N. Then
m i mj
qi − qj α ij∈A×B
≥
ij∈A×B
mi m j
1+ α2
ij∈A×B
mi mj qi − qj
2
− α2
(2.18)
For the proof of Lemma 2.4 refer to Long and Zhang [13]. In order to facilitate the reader, we repeat it. By H¨ older’s inequality, we have 2 mi mj α (2.19) mi mj ≤ m m
q − q
i j i j
qi − qj α i∈A j∈B i∈A j∈B i∈A j∈B By H¨ older’s inequality, we have mi mj qi − qj α i∈A j∈B
≤
i∈A j∈B
mi m j
2−α 2
i∈A j∈B
mi mj qi − qj 2
α/2
(2.20)
By (2.19) and (2.20) we get (2.18). Now we estimate the lower bound of f q on the collision solutions. Let SN denote the group of all the permutations of 1 N. For l = 2 N, we set ∂ l = q ∈ E ∃ s ∈ SN ∃ t¯ ∈ 0 T s.t. qs1 t¯ = · · · = qsl t¯ (2.21) First, we assume l = 2, s is the identity, and t¯ = 0. Then by the Lagrangian identity and the symmetry property qi − qj t + Tr = A 2π qi − qj t r we have f q ≥ g1 q + g2 q + g3 q
(2.22)
338
zhang and zhou
where
g1 q = r
× g2 q = r
g3 q = r
1≤i =j≤2 T/r
1 1 2 dt
q˙ − q˙j + Ma 2 i
qi − qj α
3≤i =j≤N T/r
0
m i mj
1 1 dt
q˙i − q˙j 2 + Ma 2
qi − qj α
0
(2.24)
mi m j
1≤i≤2 3≤j≤N T/r
(2.23)
mi mj
2 2M ×
0
1 2M ×
1 2M
1 1 dt
q˙i − q˙j 2 + Ma 2
qi − qj α
(2.25)
Using the estimates of the Lagrangian action integral on collision solutions of two body [7] problems we have Lemma 2.5. 2α
2−α
g1 q ≥ C1 r 2+α T 2+α where −α
C1 = AM22+α M2 =
2 i=1
1≤i =j≤2
mi m j
mi
(2.26)
(2.27) (2.28)
2α 1 1 1 A= + αa2/α+2 2π 2+α 2 2 α Let
1≤i =j≤2
B2 = mins∈SN
2 i=1
(2.29)
msi msj
msi
α/2+α
= AB C 1 2
(2.30)
(2.31)
Then 2α 2−α r 2+α T 2+α infg1 q q ∈ ∂ 2 ≥ C 1
(2.32)
noncollision periodic solutions
339
By the arguments of Degiovanni and Giannoni [7], we can get the lower bound estimate on g2 q, 2−α
g2 q ≥ C2 T 2+α
(2.33)
where −α
2+α C2 = AMN−2
MN−2 = Let
N i=3
N i=3
mi mj
(2.34)
mi
(2.35)
3≤i =j≤N msi msj BN−2 = min N α/2+α s∈SN i=3 msi
(2.36)
= AB C 2 N−2
(2.37)
Then T 2+α infg2 q q ∈ ∂ 2 ≥ C 2 2−α
(2.38)
We use inequality (2.18) of Lemma 2.4, Sundman’s inequality [19], and the arguments of Degiovanni and Giannoni [7] to estimate the lower bound for g3 q: 1+ α2 1 T 2 g3 q ≥ m m q˙ − q˙j dt + a m i mj 2M 0 1≤i≤2 3≤j≤N i j i 1≤i≤2 3≤j≤N
×
1≤i≤2 3≤j≤N
≥
1 T 2M 0 +a
mi mj qi − qj 2
m i mj
1≤i≤2 3≤j≤N
×
T 0
dt
(2.39)
1/2 2 d 2 dt m m
q − q
i j i j dt 1≤i≤2 3≤j≤N
−α/2
1≤i≤2 3≤j≤N
1+ α2
mi mj qi − qj 2
−α/2
dt
1+ α2 1 T 2 ≥ inf
r
˙ dt + a mi m j 2M 0 1≤i≤2 3≤j≤N
(2.40)
×
T
0
r −α dt r ∈ W 12 0 T R+
(2.41)
340
zhang and zhou
1 = T inf 2M
2π T
2
R +a 2
1≤i≤2 3≤j≤N
×
mi mj
1+ α2
1 R > 0 Rα
(2.42)
2 2 α+2 1+ α2 α+2 α α 2 1 2π 2 α+2 = 1+ a m i mj 2 α 2M T 1≤i≤2 3≤j≤N 2 α+2 α α 2 2−α 2 2 α+2 α − α+2 1+ = M a α+2 mi mj T 2+α 2π 2 α 1≤i≤2 3≤j≤N 2 2α α 2−α 2 1 1+ αa α+2 2π α+2 M − α+2 = mi mj T 2+α 2 α 1≤i≤2 3≤j≤N α 2−α 2−α = 2AM − α+2 mi mj T 2+α ≡ C3 T 2+α (2.43)
1≤i≤2 3≤j≤N
Let B=
mins∈SN
1≤i≤2 3≤j≤N M α/α+2
msi msj
(2.44)
= 2AB C 3
(2.45)
Then T 2−α 2+α infg3 q q ∈ ∂ 2 ≥ C 3 r inff q q ∈ ∂ 2 ≥ C 1
2α 2+α
(2.46) +C T +C 2 3
2−α 2+α
(2.47)
r 2+α + C +C T 2+α inff q q ∈ ∂ l ≥ C 1 2 3
(2.48)
It’s easy to see that for l > 2 we also have 2α
2−α
Remark. The corresponding lower bound estimate in Bessi and Coti Zelati [4] is not correct since the symmetry breaks down when they move the binary collision to the origin and they work on their domain. Lemma 2.6 [6]. Let ρ=
m i mj
sinπi − j/N α 2 πi − j σ= mi mj sin N 1≤i =j≤N
(2.49)
1≤i =j≤N
(2.50)
noncollision periodic solutions
341
Then the minimizing value for Lagrangian action f q has upper bound estimate 2 2α 2 α 2−α 1 1 1 f q ≤ + bα 2+α 2π α+2 ρ α+2 σ α+2 M −α/α+2 T 2+α 2 2 α
2−α 2+α ≡ CT
(2.51)
Now we can prove Theorem 1.1. Assume the minimizer q for f q on has a collision time t¯ ∈ 0 T . + r 2α/2+α + C Then by Lemma 2.5, we can get a lower bound estimate C 1 2 2−α/2+α T for f q, which depends on the integer r ≥ 2. By Lemma 2.6 C 3 we can choose r large enough so that 2α +C > C
r 2+α +C C 1 2 3
(2.52)
This is a contradication. Now we prove Theorem 1.2. If α = 1 m1 = m2 = m3 = 1 a = 1, and r = 2 then we have A=
3 2π2/3 4
B2 = 22/3 = 3 · 2−4/3 2π2/3 C 1
2α r 2+α C = 3 · 2−2/3 2π2/3 1
=0 C 2 B = 2 · 3−1/3 2α r 2+α C 1
= 2AB = 32/3 2π2/3 C 3 + C2 + C3 = 3 · 2−2/3 + 32/3 2π2/3
On the other hand, we compute
1 2 = √ 6 = 4 · 31/2
sinπi − j/3
3 1≤i =j≤3 π 2 9 πi − j 2 σ= = 6 sin = sin 3 3 2 1≤i =j≤3 ρ=
1/3 13 2/3 1/2 2/3 9
C= 2π 4 · 3 · 3−1/3 22 2
1 5/3 3 2π2/3 < 3 · 2−2/3 + 32/3 2π2/3 2 2α r 2+α +C =C +C =
1
2
3
342
zhang and zhou REFERENCES
1. R. Abraham and J. Marsden, “Foundations of Mechanics,” 2nd ed., Benjamin/Cummings, London, 1978. 2. V. Arnold, V. Kozlov, and A. Neishtadt, “Dynamical Systems. III. Mathematical Aspects of Classical and Celestial Mechanics,” Russian ed., 1985; English ed., Springer-Verlag, Berlin, 1988. 3. A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of three-body type, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 8 (1991), 561–649. 4. U.Bessi and V. Coti Zelati, Symmetries and noncollision closed orbits for planar N-bodytype problems, Nonlinear Anal. 16 (1991), 587–598. 5. A. Chenciner and N. Desolneux, Minima de l’int´egrale d’action et ´equilibres relatifs de n corps, C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), 1209–1212. 6. V. Coti Zelati, The periodic solutions of N-body type problems, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 7 (1990), 477–492. 7. M. Degiovanni and F. Giannoni, Dynamical systems with Newtonian type potentials, Ann. Scuola Norm. Sup. Pisa 15 (1989), 467–494. 8. G. F. Dell’Antonio, Classical solutions of a perturbed N-body system, in “Topological Nonlinear Analysis, II” (M. Matzeu and A. Vignoli, Eds.), pp. 1–86, Birkh¨auser, Basel, 1997. 9. L. Euler, De motu rectilineo trium corp¨ orum se mutuo attrahentium, Novi. Comm. Acad. Sci. Imp. Petropll. (1767), 145–151. 10. W. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), 961–971. 11. G. Hardy, J. Littlewood, and G. Polya, “Inequalities,” 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1952. 12. J. Lagrange, Essai sur le probl´eme des trois corps, 1772, Ouvres 3 (1783), 229–331. 13. Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem, Abstract, Chinese Sci. Bull. 44, No. 18 (1999), 1653–1654; Acta Math. Sinica, (English series), 16 (2000), 579–592. 14. Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations 160 (2000), 422–438. 15. K. Meyer and G. Hall, “Introduction to Hamiltonian Systems and the N-Body Problems,” Springer-Verlag, Berlin, 1992. 16. H.Pollard, “Celestial Mechanics,” Amer. Math. Soc., Providence, 1976. 17. P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157–184. 18. E. Serra and S. Terracini, Collisionless periodic solutions to some three-body problems, Arch. Rational Mech. Anal. 120 (1992), 305–325. 19. C. Siegel and J. Moser, “Lectures on Celestial Mechanics,” Springer-Verlag, Berlin, 1971. 20. A. Wintner, “Analytical Foundations of Celestial Mechanics,” Princeton Univ. Press, Princeton, NJ, 1941. 21. S. Zhang and Q. Zhou, Geometric characterization for the least Lagrangian action of N-body problems, Science in China 44 (2001), 15–20. 22. S. Zhang and Q. Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sinica., to appear.