CROSS SECTIONS IN THE THREE-BODY PROBLEM CHRISTOPHER MCCORD AND KENNETH R. MEYER
Abstract. In Dynamical Systems, Birkho gave a clear formulation of a cross section,
suggested a possible generalization to cross sections with boundary, and raised the question of whether or not such cross sections exist in the three-body problem. In this work, we explicitly develop Birkho 's notion of a generalized cross section, formulate homological necessary conditions for the existence of a cross section or generalized cross section, and show that these conditions are not satis ed in the three-body problem.
1. Statement of Results 1.1. Introduction. The existence of a global cross section to a ow places global geometric restrictions not only on the ow but on the space that underlies it. The nonexistence of a cross section suggests a certain level of complexity for the ow. Unfortunately, it is not always easy to determine whether or not a given ow admits a cross section, since the existence or non-existence depends on both the topology of the space and the dynamics of the ow. Poincare rst realized the importance of cross sections in his studies of the restricted three-body problem [20]. He was able to reduce the problem of the existence of periodic solution of the restricted problem to the problem of nding xed points of a cross section with boundary. Although Poincare was able to establish the existence of xed points in several special cases it was left to Birkho [3] to establish the general xed point theorem. Much of Birkho 's classic memoir [2] is devoted to questions about cross sections in the three body problem. Even though it is popular to assume the existence of global cross sections in dynamical systems, Reeb observed that they do not appear in classical mechanics [21]. Here is one consequence of Reeb's results. Consider a classical Hamiltonian system de ned on the cotangent bundle of a manifold Q with Hamiltonian H = K + V where K is kinetic energy (a Riemannian metric) and V : Q ! R is potential energy. Let M be the level set where H = h, then the Hamiltonian
ow de ned by H on M does not admit a compact global cross section. The applications we have in mind are to the problems of celestial mechanics and in particular to the three-body problem. In these problems the level sets are not compact in general. Also, these problems start as classical systems, but they are no longer classical systems when studied on the reduced space where all the integrals and symmetries have been eliminated. Thus Reeb's result does not apply to our studies. We are going to develop some necessary conditions for a ow on a manifold to admit a global cross section or a cross section with boundary. Then using our computations of Date : August 14, 1998. 1991 Mathematics Subject Classi cation. 34C35, 34C27, 70F07, 54H20. Key words and phrases. Birkho 's problem, cross section, three-body problem. This research partially supported by grants from the National Science Foundation and the Taft Foundation. 1
the cohomology of the integral manifolds of the spatial three-body problem in [17] and the planar three-body problem in Section 4 we discuss the existence of cross section in both senses. This discussion answers in the negative a question raised by Birkho in his classic text on dynamical systems [2]. 1.2. Global Cross Sections. Let M be a connected manifold of dimension m without boundary, : R M ! M a ow and C a submanifold of M of dimension m ; 1 without boundary. Then C is a global cross section if 1. For each point p 2 M there is a t(p) > 0 such that (t(p); p) 2 C . 2. There is a continuous function : C ! R such that (a) (t; p) 2= C for all p 2 C and 0 < t < (p). (b) ( (p); p) 2 C for all p 2 C . 3. There is an open neighborhood U of C f0g in C R such that jU is a homeomorphism from U to an open neighborhood of C in X . The function is called the return time. The function P : C ! C : p ! ( (p); p) is a homeomorphism and is called the Poincare map or rst return map. If a ow admits a cross section, it admits in nitely many. For example, if C is a cross section with return time function , and : C ! R is any continuous function, then C 0 = f((p) (p); p) j p 2 C g is also a cross section with return time function 0 (((p) (p); p)) = (p) + (P (p)) (P (p)) ; (p) (p): Clearly, h(p) = ((p) (p); p) de nes a homeomorphism h : C ! C 0 which conjugates P and P 0. But M may admit other cross sections (C 00; P 00 ) which are not conjugate. For example, on the two-dimensional torus T 2 with ow _1 = 1 _2 = 0; each Cn = f2 = n1 g, n 1 is a cross section with return map Pn(1 ) = 1 + 1=n. These are clearly not conjugate to each other. The point is, if a ow admits a cross section, then it may generate several inequivalent di eomorphisms as Poincare maps. On the other hand, suppose P~ : C~ ! C~ is a homeomorphism of a manifold C~ . Let M~ = (R C~ )= where is the equivalence relation on R C~ de ned by (t+1; p) (t; P~ (p)). De ne a ow on M~ by ~ : R M~ ! M~ : (s; [(t; p)]) ! [(t+s; p)] where [] denotes an equivalence class. ~ is called the suspension of P~ . The ow ~ admits [0 C~ ] C~ as a global cross section. Thus, every cross section on a ow produces a di eomorphism on a manifold of one dimension less; and every di eomorphism produces a ow with a cross section on a manifold of one dimension higher. If one begins with a cross section, obtains the Poincare map and constructs its suspension, the resulting manifold and ow are equivalent to the original. If one begins with a di eomorphism and constructs its suspension, then any slice Ct = ft = t0g will be a section with return map conjugate to the original di eomorphism (though other, non-conjugate return maps may also exist). The rst and fundamental question for cross sections is existence: given a ow on a manifold M , does there exist a cross section? In particular, are there computable necessary or sucient conditions for the existence of a cross section? A related but distinct problem is to classify all conjugacy classes of cross sections and Poincare maps. While this second question has received considerable attention [10, 11, 29], the rst has not received as much 2
attention as one might expect. Some simple necessary conditions for the existence of a global cross section are formulated in the following: Theorem 1.1. If the ow : R M ! M on the manifold M admit a global cross section C , then M is a ber bundle over S 1 with ber C . There is a long exact homology sequence ;P ! Hk+1(M ) ! Hk (C ) id;! Hk (C ) ! Hk (M ) !
If C is of nite type, there exists an integer polynomial Q(t) with 0 Q(t) PC (t)
such that PM (t) = (1 + t)Q(t). If C is of nite type, then (M ) = 0, (the Euler characteristic of M is zero). If C is of nite type, H1(M ; Z) has a factor Z. The ow has no equilibrium points. Remark. Of nite type means that the homology of M is nitely generated. In that setting, PM (t) is the Poincare polynomial of M : a formal polynomial whose nth coecient is the nth Betti number of M . The Euler characteristic is the alternating sum of the Betti numbers, and is also the value of the Poincare polynomial evaluated at t = ;1. The inequality 0 Q(t) PC (t) should be interpreted term-by-term: each coecient qn is non-negative, and less than or equal to the nth Betti number of C .
1.3. General Cross Sections. Reeb's theorem indicates that global cross sections are not common. Theorem 1.1 shows that, among other things, no ow with an equilibrium point can admit a global cross section. But there are situations in which a global cross section \almost" exists. As the simplest possible example, consider the ow _ = 1 r_ = 0 on R 2 where (r; ) are polar coordinates. Let C be a closed ray emanating from the origin. Then C n @C is a cross-section for the ow on R2 n f0g. Now let M and be as above but now C is a submanifold of M of dimension m ; 1 with boundary @C of dimension m ; 2. Let int C = C n @C be the interior of C . Then C is a cross section with boundary if 1. The boundary @C of C is invariant under the ow . 2. C n @C is a cross section for the ow on X n @C . 3. The return time and Poincare map on C n @C extend continuously to @C . Reeb's Theorem says that global cross sections in an energy surface do not exist for classical dynamical systems but cross sections with boundary often exist. Generically on a compact manifold, the Hamiltonian has a nondegenerate minimum of general elliptic type [15] and for a two degree of freedom system the ow near such an equilibrium point admits a cross section with boundary on an energy surface. We will illustrate this with a simpli ed example. Consider a Hamiltonian system on R 4 with Hamiltonian H = !21 (x21 + y12) + !22 (x22 + y22) 3
were !1; !2 > 0 are constants (frequencies). H has a minium at the origin and H = h > 0 is an ellipsoid homeomorphic to S 3. Change to action-angle coordinates I1 ; I2; 1 ; 2 by Ii = 21 (x2i + yi2); i = arctan yi=xi so that H = !1I1 + !2I2 and the equations of motion become _ 1 = ;!1 ; I_1 = 0; _ 2 = ;!2 : I_2 = 0; A geometric model for S 3 can be obtained from these coordinates. Consider the set where H = !1 > 0 which we will call S 3 . Since !1 I1 + !2I2 = !1 we can ignore the I2 coordinate and use I1 ; 1; 2 as coordinates on S 3 but remember that 0 I1 1 and 1 ; 2 are angles de ned modulo 2. The closed unit disk is coordinatized by (I1 ; 1); 0 I1 1 using the usual conventions of action-angle (polar) coordinates. For each point in the open unit disk, there is a circle with coordinate 2 (de ned modulo 2), but when I1 = 1; I2 = 0; so the circle collapses to a point over the boundary of the disk. Thus a geometric model for S 3 is two solid cones with points on the boundary identi ed as show in Figure 1. There are always two periodic orbits namely where I1 = 0 and I1 = 1 and these would be called the normal modes by engineers. The closed disk where 2 = 0 mod 2 is a cross section with boundary, since its boundary is the periodic orbit where I1 = 1 and all other solutions cross the open disk ( _2 = ;!2 6= 0). The return time is 2=!2. This cross section is shaded in Figure 2. There is another cross section which is an annulus with both the periodic orbits as boundaries. This cross section is de ned by 1 + 2 = 0 mod 2 with return time 2=(!1 + !2). Conley [7] showed that the ow near a primary in the restricted three-body problem after regularization of collisions is but a perturbation of this examples and was able to use this observation to invoke Birkho 's xed point theorem to establish the existence of long period periodic-solutions. The same basic existence and uniqueness questions arise for cross sections with boundary. The natural generalization of Theorem 1.1 gives some veri able necessary conditions for the existence of a cross section with boundary. Theorem 1.2. If the ow : R M ! M on the manifold M admits a cross section with boundary C , then M n @C is a ber bundle over S 1 with ber C n @C .
There is a long exact homology sequence ;P ! Hk+1(M; @C ) ! Hk (C; @C ) id;! Hk (C; @C ) ! Hk (M; @C ) ! If C and @C are of nite type, then there exists a polynomial Q(t) with ; minfP@C (t); tP(C;@C ) (t)g Q(t) P(C;@C )(t) such that PM (t) ; P@C (t) = (1 + t)Q(t). If C and @C are of nite type, then (M ) = (@C ). All equilibrium points of the ow must lie in @C .
This theorem will be proved in Section 2.
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Figure 1. Model of S 3
Figure 2. Orbits on S 3
1.4. Applications to the Three-Body Problem. Here we will apply the above results on cross sections to the three-body problem. The three-body problem is a system of di erential equations describing the motion of three mass points moving in a Newtonian inertial frame under the in uence of their mutual gravitational attraction. Let the particles have masses m1 ; m2; m3 ; positions u1; u2; u3; and velocities v1; v2 ; v3 respectively. The masses are positive constants and the positions and velocities are two or three dimensional vectors depending on whether we are discussing the planar or the spatial problem. 5
Written as a system of rst order equations the equations of motion are u_ i = vi; (1) @U ; i = 1; 2; 3; miv_ i = @u i where the dot represents the derivative with respect to time, : = d=dt, U is the self-potential X Gmi mj ; (2) U= 1i 0; k 2 intkR (c; h) < S3 ; 1 ; 1 > 0; k 2 @ kR (c; h) ! (k) = : 3 B < 0; k 2 intkR (c; h) This, combined with the information about kR (c; h) and c(c; h) encoded in Proposition 3.3 and Figure 4, will enable us to establish the homotopy types of m(c; h) and mR (c; h), and so determine their homology. 4. The Homology of the Integral Manifolds With this analysis in hand, it is now a relatively simple matter to determine the homotopy types of the Hill's regions and integral manifolds. The rst step is to observe that each of these open manifolds has a strong deformation retraction onto a lower dimensional compact subcomplex. Proposition 4.1. For each range of , there is a 1-complex l(c; h) kR(c; h) such that kR (c; h) has a strong deformation retraction onto l(c; h) and mR (c; h) has a strong deformation retraction onto ;1 ! ;1 (l(c; h)). The sets l(c; h) are shown in Figure 5 Proof. First, it is clear that each kR (c; h) has a strong deformation retraction : kR (c; h) [0; 1] ! kR (c; h) onto lR (c; h). The only restriction to be observed in constructing is to require @ kR (c; h) [0; 1] ;1 (@ kR (c; h)) (@ kR (c; h) [0; 1])) [ (kR(c; h) f1g) : This lifts to a strong deformation retraction ~ : mR (c; h)[0; 1] ! mR (c; h) with ~1 (mR (c; h)) ; 1!;1 (l(c; h)). To see that such a lift exists, let A = ;1 (intkR(c; h)). By our choice of , A is dense in kR (c; h) [0; 1], and over both B and (A), the projection ! is a bration. The homotopy lifting condition guarantees the existence of a lift ~ over A. On the complement of A, maps into @ kR (c; h), and ! is one-to-one there. Thus, the unique continuous extension to all of kR (c; h) [0; 1] is de ned by setting ~ = (! );1 ! 15
Figure 5. The 1-complex l(c; h)
on the complement of A. With this, we can now determine the homotopy types of the integral manifolds, Hill's regions and reduced spaces. The homological values of Table 2 follow immediately. Theorem 4.1. The homtopy types of m(c; h), mR (c; h), h(c; h) and hR(c; h) are given in the following table: Case m(c; h) mR (c; h) h(c; h) hR (c; h) 1 1 1 1 1 1 1 1 i S (S _ S ) S _S S (S _ S ) S1 _ S1 ii S 1 WS 3 (S 1 _WS 1) S 3 (S 1 _WS 1) S 1 (SW1 _ S 1) SW1 _ S 1 W 1 4 1 4 1 1 1 vii S ((W3 S ) _ (W4 S )) (W3 S ) _ (W4 S ) S (W4 S ) S1 4 W viii S 1 (( 2 S 4 ) _ ( 3 S 1 )) ( 2 S 4 ) _ ( 3 S 1) S 1 ( 3 S 1) S1 3 ix S 1 ((S 4 _ S 1) t S 1) (S 4 _ S 1) t S 1 S 1 ((S 1 _ S 1) t S 1) (S 1 _ S 1) t S 1 1 1 1 1 x S (S t S t S ) S1 t S1 t S1 S 1 (S 1 t S 1 t S 1) S 1 t S 1 t S 1 Proof. Combining Propositions 3.2 and retraction, it is clear that hR (c; h) is homotopic to the wedge of circles l(c; h). Similarly, the homotopy type of mR (c; h) is obtained by combining the results of Corollary 3.1 and Proposition 4.1. In all cases, lR (c; h) consists of arcs that either lie entirely in @ kR (c; h), entirely in intkR (c; h), or lie in intkR (c; h) with endpoints in @ kR (c; h). In the rst case, the ber in mR (c; h) is itself an arc; in the second case it is S 3 I ; while in the third case it is S 4 (i.e. S 3 I , with the ends collapsed to points). In cases i and ii, the 1-complex lR(c; h) lies entirely in intkR (c; h), so its preimage in mR (c; h) is simply B 3 lR (c; h) in case i, and S 3 lR (c; h) in case ii. In case x, lR (c; h) lies entirely in @ kR (c; h), and so is homeomorphic to its preimage. In the remaining cases, the arcs in lR (c; h) with endpoints in @ kR (c; h) e ectively attach a 4-sphere at two points to @ kR (c; h). This is homotopic to attaching the wedge product of an arc and a 4-sphere. 16
The homotopy types of h(c; h) and m(c; h) are simply the products of S 1 with h(c; h) and m(c; h) respectively. A homological calculation suces to justify this. Namely, since H 2 (mR (c; h)) = H 2 (hR (c; h)) = 0 for all , the Thom classes of all of the S 1-bundles S 1 ! m(c; h) ! mR (c; h) S 1 ! h(c; h) ! hR (c; h): must be trivial. Since the Thom class determines the homotopy type of the total space, m(c; h) and h(c; h) must have the homotopy types of the trivial (i.e. product) bundles. As a nal note, we look more carefully at the reduced integral manifold in case x. In this case, kR(c; h) consists of three half-open annuli. For each half-open interval, the end-point lies in @ kR (c; h) (and hence has a single point as its preimage in mR (c; h)), while all other points have S 3 as their preimage. The preimage of the entire interval is the cone on S 3, or D4. Thus we have: Proposition 4.2. For > 9, mR (c; h) is homeomorphic to three disjoint copies of D4 S 1 . Appendix: The Spatial Three-Body Problem for Positive Energy In this appendix, we correct the erroneous computation of the homology of H (MR(c; h))
for < 0, that was presented in [17]. We use the notation and approach of [17]. There, it was shown that the integral manifold MR (c; h) could be understood through a series of projections ! MR (c; h) ! HR (c; h) ! KR (c; h) ! C(c; h): For positive energy, the set C(c; h) is a triangle with the three vertices deleted, while ;1(c) is a 2-sphere for c in the interior of the triangle and a point for c on one of the three boundary lines. Over each point k 2 K, the ber ;1 !;1(k) is contractible. The integral manifold MR(c; h) is thus homotopic to KR (c; h). This is in turn homotopic to a singular ber bundle over a Y (i.e. a 1-complex with three edges all meeting at a single common vertex), with the ber over each of the three end-points a point and the ber over all other points a 2-sphere. The problem is to correctly identify the limiting behavior of the S 2 bers as points in the interior of the Y approach the boundary. It was this step that was incorrectly described in [17]. A point in C records the shape of the triangle formed by the three masses, while its preimage in KR describes the orientation of the triangle relative to the angular momentum vector. The points in @ C are the collinear con gurations. In the presence of non-zero angular momentum, collinear con gurations must lie in the invariant plane orthogonal to the angular momentum vector. Thus the noncollinear con gurations that limit onto collinear must also lie in or asymptotically approach the invariant plane. That is, as points c 2 int(C) approach c0 2 @ C, it is not the entire 2-sphere ;1 (c) that limits onto the single point ;1(c0), but just the equator. Let A = ;1 (int(C)) and B be a small neighborhood of ;1 (@ C). Then A ' S 2, each of the three components of B is contractible, and each of the three components of A \ B is homotopic to a circle. The Mayer-Vietoris sequence is then 0 ! Z ! H2(MR ) ! Z3 ! 0 ! H1(M) ! Z3 ! Z4 ! H0 (MR); from which the values for Case I in Table 1 follow. In fact, for < 0, it is not hard to identify the homotopy type of MR(c; h) ' KR(c; h). The ber in KR (c; h) over each of the arms of 17
the Y is homotopic to a 2-disk, so the entire space has the homotopy type of a 2-sphere with three disks sewn onto the equator, which is homotopic to a wedge of four 2-spheres. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
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University of Cincinnati, Cincinnati, Ohio 45221-0025 E-mail address :
[email protected]
University of Cincinnati, Cincinnati, Ohio 45221-0025 E-mail address :
[email protected]
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