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1 Journal of Differential Equations 96 (992), Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge, Massachusetts 0238 Abstract 2 2 [0] McGehee[7]. Sitnikov[3] Alekseev[] Sitnikov Moser[9] [3,8,2] 2 Smale Sitnikov [0] 2 McGehee[7] Easton[2] Robinson[]

2 2 Llibre and Simó[5,6] J.Llibre [5] Llibre and Simó C ± 2 [5,6] K.Meyer 2. R 2 3 P,P 2,P 3 P P 2 μ μ 0 μ P 3 P 3 P P 2 P P 2 P P 2 q =(q,q 2 ) P 3 p =(p,p 2 )=(q,q 2) P 3 P 3 q = p, () p = U q. U U q q U U = μ (q μx 2 ) 2 +(q 2 μy 2 ) + μ (q 2 +( μ)x 2 ) 2 +(q 2 +( μ)y 2 ). 2 x 2,y 2 P P 2 x y x 2 =cost, y 2 =sint. μ =0 μ μ U 2

3 U = ( + μ q 2 + q2 2 (q 2 + q2) + q cos t + q 2 sin t 2 /2 (q 2 + q2) 2 3/2 ) + + O(μ 2 ). ((q +cost) 2 +(q 2 +sint) 2 ) /2 μ =0 t t μ q = x 2 S s q = x 2 s p = ys + x 2 ρis. is s ρ P 3 U = x 2 + μx 2 ( +x 2 cos(t θ)+ (2) ) + O(μ 2 ), (3) ( + 2x 2 cos(t θ)+x 4 ) 3/2 x = 2 x3 y, y = x 4 + x 6 ρ 2 + μg (x, t θ)+o(μ 2 ), θ = x 4 ρ, ρ = μg 2 (x, t θ)+o(μ 2 ). (4) θ S s =(cosθ, sin θ) g (x, t θ),g 2 (x, t θ) g (x, t θ) =x ( 4 2x 2 +x 2 ) cos(t θ) cos(t θ), ( ( + 2x 2 cos(t θ)+x ) 4 ) 3/2 (5) g 2 (x, t θ) =x 4 sin(t θ). ( + 2x 2 cos(t θ)+x 4 ) 3/2 5 2 y2 + 2 x4 ρ 2 U ρ = C, 3

4 C ρ ρ μx 2 ( ) ρ = ρ 0 +x 2 cos(t θ)+ + O(μ 2 ), ( x 4 ρ 0 ) ( + 2x 2 cos(t θ)+x 4 ) 3/2 ρ 0 = ± (6) x 4 (y 2 2x 2 2C). x 4 ± P 3 t t θ s s = t θ, s S. x = 2 x3 y, y = x 4 + x 6 ρ 2 + μg (x, s)+o(μ 2 ), s = x 4 ρ. ρ x, y, s C (6) μ =0 x = 2 x3 y, y = x 4 + x 6 ρ 2, (8) s = x 4 ρ. ρ (ρ P 3 ) x, y s H x y H(x, y, ρ) = 2 y2 + 2 x4 ρ 2 x 2. (9) (7) Figure H(x, y, ρ) (H P 3 ) Figure 4

5 x, y (x >0 ) H (0, 0) x 3 x(t) =ξ(t, C) = 2 (3t + 9t 2 + C 6 ) 2/3 +(3t 9t 2 + C 6 ) 2/3 C 2, ± 2ξ 2 (t) ξ 4 (t)c 2 for x 0, y(t) =η(t, C) = 2ξ 2 (t) ξ 4 (t)c 2 for x 0. (0) ± C ξ(t, C) t η(t, C) t t 0 R (ξ(t t 0,C),η(t t 0,C)) ξ(t, C) η(t, C) s s x 0 (s) y 0 (s) s x 0 (s) =ξ[t(s),c] y 0 (s) =η[t(s),c] θ 0 (?) x 0 (s) s y 0 (s) 0 <μ x y s dx ds = 2 x3 y x 4 ρ, dy ds = x4 + x 6 ρ 2 + μg (x, s) x 4 ρ x 4 ρ + O(μ2 ). ρ x, y, s C (6) s x 4 ρ 0x ρ<0 μ 0 s 2π t ω t α x 3 ω α McGehee[7] ω α. () {x 0} γ : x =0,y =0,s S x>0 W s (γ) {x>0} W u (γ) {x>0} C ω α () 5

6 McGehee[7] Robinson[] McGehee Robinson[] 2 () C μ, μ 0 (μ =0 ) 3. μ =0 W s (γ) W u (γ) μ 0 P 3 t =0 x ω α γ Σ s 0 s = s 0 (x u μ (s, s 0),yμ u(s, s 0)) W u (γ) (x u μ(0,s 0 ),yμ(0,s u 0 )) Σ s 0 (x s μ(s, s 0 ),yμ(s, s s 0 )) W s (γ) (x s μ (0,s 0),yμ s(0,s 0)) Σ s 0. C > 2 C x s μ (s, s 0)=x 0 (s s 0 )+μx s (s, s 0)+O(μ 2 ), s [s 0, ), yμ s(s, s 0)=y 0 (s s 0 )+μy s(s, s 0)+O(μ 2 ), s [s 0, ), x u μ (s, s 0)=x 0 (s s 0 )+μx u (s, s 0)+O(μ 2 ), s (,s 0 ], yμ u(s, s 0)=y 0 (s s 0 )+μy u(s, s 0)+O(μ 2 ), s (,s 0 ]. (2) x s (s, s 0 ),y s (s, s 0 ) x u (s, s 0 ),y u (s, s 0 ) (x 0,y 0 ) () 2 () Gronwall (x 0 (0),y 0 (0)) O(μ) (x 0 (s),y 0 (s)) O(μ) (2) W u (γ),w s (γ) C C μ>0 C 2 C > 2 6

7 H(x, y) = 2 y2 + 2 x4 ρ 2 x 2 (9) ρ (6) μ 0 H dh ds = μ(yg (x, s)+x 4 ρg 2 (x, s)) x 4 + O(μ 2 ). (3) ρ d(s 0 ) d(s 0 )=(x s μ (s 0,s 0 ),y s μ (s 0,s 0 )) (x u μ (s 0,s 0 ),y u μ (s 0,s 0 )). W s (γ) W u (γ) Σ s 0 s = s 0 d(s 0 ) W s (γ) W u (γ) H N (x 0 (0),y 0 (0)) (x 0 (0),y 0 (0)) H(x, y) H N (x 0 (0),y 0 (0)) = (H x (x 0 (0),y 0 (0)),H y (x 0 (0),y 0 (0))). d(s 0 ) (x 0 (0),y 0 (0)) H(x, y) d(s 0 ) d(s 0 ) = HN (x 0 (0),y 0 (0)) H N (x 0 (0),y 0 (0)) (xs μ(s 0,s 0 ) x u μ(s 0,s 0 ),y s μ(s 0,s 0 ) y u μ(s 0,s 0 )) = μ HN (x 0 (0),y 0 (0)) H N (x 0 (0),y 0 (0)) (xs (s 0,s 0 ) x u (s 0,s 0 ),y s (s 0,s 0 ) y u (s 0,s 0 )) + O(μ 2 ). x s (s, s 0),y s (s, s 0),x u (s, s 0),y u (s, s 0) d(s 0 ) Σ s 0 (x 0 (0),y 0 (0)) W u (γ),w s (γ) H(x s μ (s 0,s 0 ), y s μ (s 0,s 0 )) = H(x 0 (0),y 0 (0)) + μh N (x 0 (0),y 0 (0)) (x s (s 0,s 0 ),y s (s 0,s 0 )) + O(μ 2 ), H(x u μ (s 0,s 0 ), y u μ (s 0,s 0 )) = H(x 0 (0),y 0 (0)) + μh N (x 0 (0),y 0 (0)) (x u (s 0,s 0 ),y u (s 0,s 0 )) + O(μ 2 ). d(s 0 ) H N (x 0 (0),y 0 (0)) = H(x s μ (s 0,s 0 ),yμ s(s 0,s 0 )) H(x u μ (s 0,s 0 ),yμ u(s 0,s 0 )) + O(μ 2 ) dh(x, y) = ds + O(μ 2 ) ds = μm(s 0 )+O(μ 2 ), M(s 0 ) dh M(s 0 ) = μ ds (x 0(s s 0 ),y 0 (s s 0 ))ds yg (x 0 (s s 0 ),s)+x 4 = 0(s s 0 )ρg 2 (x 0 (s s 0 ),s) ds. x 4 0(s s 0 )ρ (4) (5) 7

8 ρ = C d(s 0 )= M(s 0 ) μ H N (x 0 (0),y 0 (0)) + O(μ2 ) = M(s 0 ) μ 2x 3 0(0)C 2 2x 0 (0) + O(μ2 ) (6) = μ C 2 2 M(s 0)+O(μ 2 ). g (x, s)+x 4 ρg 2 (x, s) x 4 ρ = g 2(x, s) x 4 ρ + d ( ( x 2 +x 2 cos(t θ) ds )) + + O(μ 2 ). ( + 2x 2 cos(t θ)+x 4 ) 3/2 ρ ρ 0 x(s),y(s) s M(s 0 )= = = = g 2 (x(s s 0 ),s) ds x 4 0(s s 0 )ρ 0 g 2 (x(s),s+ s 0 ) ds x 4 0(s)ρ 0 g 2 (x 0 (s),s+ s 0 )dt ( x 4 0 (s)sin(s + s 0) ) dt. ( + 2x 2 0(s)cos(s + s 0 )+x 4 0) 3/2 (7) s 0 = π M(s 0 )=0 s 0 = π M(s 0 ) M (s 0 ) s0 =π 0 M (π) = x 4 0 (s)cos(s + π)dt x 4 0 (s)cos(s + π) ( + 2x 2 0(s)cos(s + π)+x 4 0(s)) 3/2 dt 3x 6 0 sin 2 (s + π) dt. ( + 2x 2 0(s)cos(s + π)+x 4 5/2 0(s)) (8) x 0 (s) =ξ(t, C) (0) s s = t θ = t t 0 x 4 ρdt = t + t 0 ξ 4 (t, C)Cdt. C = ± 2 M(s 0 ) M (π) t =0 t =0 C = ± 2 +2ξ 2 cos(s + π) +ξ 4 =0 +2ξ 2 cos(s + π) +ξ 4 t =0 C > 2 C ± 2 8

9 M (0) C > 2 C 2 M (π) 0 C μ γ W s (γ) W u (γ) μ W s (γ) W u (γ) μ s 0 = π Σ s 0 W s (γ) x W u (γ) C μ W u (γ) W s (γ) ( C ) ([4] ) W u (γ) W s (γ) μ C x p(μ, C) k(μ, C) p(μ, C) W s (γ) p(μ, C) W u (γ) k(μ, C) k(μ, C) 0 p(μ, C) W u (γ) W s (γ) W u (γ) W s (γ) C μ k(μ, C) k(μ, C) 0 ( W u (γ) x ) C μ k(μ, C) 0 C μ k(μ, C) C μ μ C > 2 W u (γ) W s (γ) k(μ, C) 0 k(μ, C) C μ μ C k(μ, C) 0 k(μ, C) C μ W u (γ) W s (γ) C = C C μ k(μ, C ) k(μ, C ) 0 k(μ, C) μ [0, ] k(μ, C ) k(μ, C ) 0 2. μ γ W s (γ) W u (γ) 4. Smale-Birkhoff ( s 9

10 ) t ± lim sup r = lim inf r< r P 3 r = q 2 + q2 2 = x 2 t ± r t lim sup r< Moser[9] s 0 Γ=Σ s 0 C Γ p R 2 W s (p) W u (p) (Fig.2 ) φ q R k = k(q) φ k (q) R ( ) k>0 q R D φ(q) =φ k (q) for all q D. Moser φ R φ D (shift) D Fig.2 3. φ I D S = N Z ( ) I φ S τ τ : S I σ S φτ = τσ. [9] Smale-Birkhoff 0

11 q I k(q) 4. I 3 q I S I s = {...,s 3,s 2,s,s 0,s,s 2,s 3,...) sup{s i,i Z} = Fig.2 R R 5. (989)[0] 2 μ 5. μ [0, ] P P 2 μ μ References. V.M. Alekseev, Quasirandom dynamical systems, I, II, III, Math. USSR-Sb. 5 (968), 73-28; 6 (968), ; 7 (969), R. Easton, Parabolic orbits for the planar three-body problem, J. Differential Equations 52 (984), R. Easton and R. McGehee, Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere, Indiana Univ. Math. J. 28 (979), J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, J. Llibre and C. Simó, Some homoclinic phenomena in the three-body problem, J. Differential Equations 37 (980), J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann. 248 (980),

12 7. R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations 4 (973), R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal. 5 (984), J. Moser, Stable and Random Motions in Dynamical Systems, Annals of mathematics Studies, No.77, Princeton Univ. Press, Princeton, NJ, H. Poincaré, Les méthodes nouvelles de la mécanique céleste III, Gauthier-Villars, Paris, C. Robinson, Homoclinic orbits and oscillation for the planar three-body problem, J. Differential Equations 52 (984), D. Saari and Z. Xia, The existence of oscillatory and superhyperbolic motion in Newtonian systems, J. Differential Equations 82 (989), K. Sitnikov, The existence of oscillatory motion in the three-body problem, Dokl. Akad. Nauk USSR 33 (960), A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. of Math. 98 (973),

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