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4 On the Integrability of the n–Centre Problem 0 0 2 Andreas Knauf Iskander A. Taimanov n ∗ † a J 6 1 Abstract ] S It is known that for n 3 centres and positive energies the n- D ≥ centre problem of celestial mechanics leads to a flow with a strange . h repellor and positive topological entropy. t a Here we consider the energies above some threshold and show: m Whereas for arbitrary g > 1 independent integrals of Gevrey class g [ exist, no real-analytic (that is, Gevrey class 1) independent integral 1 exists. v 2 0 1 Introduction 2 1 0 In [BT] the existence of a smoothly integrable geodesic flow on a compact 4 0 manifold with positive topological entropy was established. Positivity of / h topological entropy is seen as an indication of complex dynamics, whereas t a integrability of a Hamiltonian flow is a metaphor for its simplicity. So coex- m istence of these two properties may not have been expected. In fact, as we v: show here, such a coexistence takes place in natural physical problems. i To be more specific, in this note we consider the n–centre problem of X celestial mechanics in d = 2 and 3 dimensions. We denote by ~s Rd, Z r k k a R 0 the location resp. strength of the k–th centre, assuming∈~s = ~s fo∈r k l \{ } 6 k = l. Then the Hamiltonian function 6 Hˆ : T∗Mˆ R , Hˆ (p~,~q) = 1~p2 +V(~q), → 2 ∗Mathematisches Institut, Universität Erlangen-Nu¨rnberg, Bismarckstr. 11, D–91054 2 Erlangen, Germany. e-mail: [email protected] †Institute ofMathematics,630090Novosibirsk,Russia. e-mail: [email protected] 1 with potential n Z V : Mˆ R , V(~q) = k , (1) → − ~q ~s k k=1 k − k X on the cotangent bundle T∗Mˆ of configuration space Mˆ := Rd ~s ,...,~s 1 n \{ } generates a – in general incomplete – flow. Denoting by ωˆ := d dq dp ↾ the restricted canonical symplectic i=1 i∧ i T∗Mˆ form, there exists a unique smooth extension P (P,ω,H) of the hamiltonian system (T∗Mˆ ,ωˆ,Hˆ ) (2) such that the flow Φ : R P P of H is complete (see [Kn], Thm. 5.1). × → Concerning integrability of the flow, the following is known: For n = 1 this Hamiltonian system is integrable, with angular momen- • tum (~q ~s ) p~ , x = (p~,~q), ~q =~s L~ : P R3 , L~(x) = − 1 × 6 1 → 0 , otherwise (cid:26) for dimension d = 3 being a real analytic constant of motion. For Z > 0 this is called the Kepler problem. 1 For n = 2 centres one introduces elliptic prolate coordinates to analyt- • ically integrate the flow Φ, see e.g. [Ar] or [Th]. For n 3 centres and d = 2 Bolotin showed in [Bo] the nonexistence of • ≥ an analytic integral of the motion which is non–constant on an energy shell H−1(E), E > 0, see also the discussion in Fomenko [Fo]. For d = 3 and a collinear configuration of centres the angular momen- • tum w.r.t. that axis is an additional constant of the motion, independent of the number n of centres. However, for d = 3 it was proved by the first author [Kn] for sufficiently large energies E > E and by th Bolotin and Negrini [BN, BN2] for nonnegative energies E 0 that ≥ the topological entropy of the flow, restricted to the set b of bounded E 2 orbits on H−1(E) is positive (and h (E) = 0 for b = ). Further- top E ∅ more h (E) is zero for n = 1 and 2, and h (E) > 0 if n 3 and all top top ≥ centres being attracting or not more than two ~s being on a line (for k collinear configurations with Z ,...,Z < 0 one has h (E) = 0 for 1 n top E > 0). In the present paper it is proved that for d = 2, attracting centres (Z > 0) and E > 0, k • for d = 3, arbitrary Z = 0, non-collinear configurations of centres and k • 6 E > E where the threshold energy level E is determined by the th th data Z , ~s , k = 1,...,n, k k the n-centre problem restricted onto the energy level E admits d 1 indepen- − dent integrals of motion which are smooth (and moreover are of the Gevrey class g for any g > 1) (see Thm. 1). Therefore the restricted problem is smoothly integrable. On the other hand if the affine span of the (non-collinear) centres equals R3 then the restricted problem is not integrable in the real-analytic sense, that is Gevrey class 1 (see Thm. 2). The article is based on the analysis of the n-centre problem given in the paper by the first author [Kn] where, in particular, it is shown that for high energies there are maps which relate scattering orbits to their asymptotics given by scattering orbits of the Kepler problem which is integrable (the necessary facts extracted from [Kn] are exposed in sections 2–3 of the present paper). Such a relation of the n-centre problem with an integrable problem leadstothesmoothintegrabilityofthen-centreproblemwhichisprovedhere. Here we use a trick for constructing smooth first integrals from discontinuous preserved quantities similar to others used in [Bu, BT] We would like to notice that in [Bo] the nonexistence of an additional analytic integral of motion for the two-dimensional problem was derived by Bolotin from Kozlov’s theorem [Ko] which reads that the geodesic flow of (real)-analytic metric on a closed oriented surface of genus g > 1 does not admit an additional analytic first integral. Although one can expect that the integrability of a problem by analytic integrals of motion implies vanishing of the topological entropy it is not proved until now. Therefore the results from [BN, Kn] do not imply the nonexistence ofacompletefamilyofanalyticfirst integrals. Weprovethatby 3 ananalysis of the set formed by bounded orbits which supports the restricted flow with positive topological entropy. 2 Known Smoothness Results There are three basic types of motion: bounded, scattering and trapped, corresponding to the disjoint subsets b,s,t P with ⊂ b± := x P ~q( R+,x) is bounded , b := b+ b−, { ∈ | ± } ∩ s± := x P x b± and H(x) > 0 , s := s+ s− { ∈ | 6∈ } ∩ andt := s+ s−. The orbits in s go tospatial infinity inbothtime directions, △ but because of the long range character of the effective potential of strength Z := n Z , one describes these limits by comparison with (regularized) ∞ k=1 k Kepler flow Φt : P P generated by the extension of P ∞ ∞ → ∞ Z Hˆ : T∗(Rd 0 ) R , Hˆ (p~,~q) := 1~p2 ∞ (3) ∞ \{ } → ∞ 2 − ~q k k Identifying the two phase spaces P and P outside a region projecting to a ∞ ball in configuration space which contains all singularities, the Møller transformations Ω± : P := x P H (x) > 0 s± , Ω± := lim Φ−t Id Φt (4) ∞,+ { ∈ ∞ | ∞ } → t→±∞ ◦ ◦ ∞ exist as pointwise limits, and are measure-preserving diffeomorphisms (Thm. 6,3 and 6.5 of [Kn]). Similarly the asymptotic limits of the momentum ~p± : s± Rd , ~p±(x ) := lim ~p Φt(x ) (5) 0 0 → t→±∞ ◦ are smooth. Finally we define time delay τ : s R of a scattering state → x s by ∈ τ(x) := lim σ(R) Φt(x) (6) R→∞ R ◦ − Z (cid:16) 1(σ (R) Φt Ω+(x)+σ (R) Φt Ω−(x)) dt, 2 ∞ ◦ ∞ ◦ ∗ ∞ ◦ ∞ ◦ ∗ (cid:17) where σ(R) : P 0,1 and σ (R) : P 0,1 are the characteristic ∞ ∞ → { } → { } functions σ(R)(x) := θ(R ~q(x) ) and similarly for σ (R). That asymp- ∞ −k k totic difference between the time spent by the orbit and its Kepler limits inside a ball of large radius diverges near b t. However τ is smooth, as the ∪ Møller transformations are. 4 3 Analyticity Properties Proposition 1 For a potential V of the form (1) the following maps are real-analytic: 1. the flow Φ : R P P × → 2. the Møller transformations Ω± : P s± and asymptotic momenta ∞,+ → ~p± : s± Rd → 3. the time delay τ : s R. → Proof. 1) We first indicate the definition of phase space P in order to show that for the potential (1) the smooth extension (2) actually works in the real-analytic category (in [Kn] more general non-analytic potentials V were considered). We assume d = 3, the case of d = 2 dimensions following by restriction. For small ε > 0 in the phase space neighbourhood 3 Z Uˆε := (p~,~q) T∗Mˆ ~q ~s < ε, p~ 2 > l . (7) l ∈ k − lk | | 2 ~q ~s (cid:26) (cid:12) k − lk(cid:27) (cid:12) (cid:12) of the lth centre the following real-analytic coordinates are used: (cid:12) The restriction of the Hamiltonian function, which splits into • Z Hˆ(p~,~q) = Hˆ (p~,~q)+W (~q) with Hˆ (p~,~q) := 1~p2 l (8) l l l 2 − ~q ~s l k − k and Z i W (~q) = − . l ~q ~s i i6=l k − k X The angular momentum • Lˆ : Uˆε R3 , Lˆ (p~,~q) := (~q ~s ) p~. (9) l l → l − l × relative to the position ~s . l 5 The time Tˆ : Uˆε R after which the Kepler orbit generated by Hˆ is • l l → l in its pericentre w.r.t. ~s l kq~−~slk rdr ˆ T (p~,~q) := sign((~q ~s ) p~). l l Zrmin(p~,q~) 2r2Hˆl(p~,~q)+2Zlr−Lˆ2l(p~,~q) · − · q (10) The neighbourhood Uˆε T∗Mˆ is defined in a way which makes the l ⊂ pericentre unique. Its distance from ~s equals l −Zl+√Zl2+2Hˆl(p~,q~)Lˆ2l(p~,q~) ,Hˆ = 0 rmin(p~,~q) := 2Hˆl(p~,q~) l 6 . (11) ( Lˆ2(p~,~q)/2Z ,Hˆ = 0 l l l The Runge-Lenz vector relative to ~s is given by l • ~q ~s F~ : Uˆε R3 , F~ (p~,~q) := ~p Lˆ (p~,~q) Z − l . l l → l × l − l ~q ~s l k − k Since F~ 2 > Z2/4 > 0, the pericentral direction | l| l Fˆ : Uˆε S2 , Fˆ := F~ / F~ (12) l l → l l | l| is well-defined. As Lˆ Fˆ = 0, we get a real-analytic diffeomorphism onto the image l l · ˆ : Uˆε T∗(R S2) ¯0, (p~,~q) (Tˆ,Lˆ ;Hˆ ,Fˆ) Y l → × \ 7→ l l l l ontoapuncturedneighbourhoodofthezerosection¯0ofthecotangentbundle. The flow generated by Hˆ is linearized in these coordinates, and extended on l Uε := Uˆε (R S2) by l l ∪ × (0,0;h,f) (h,f) R S2 := (T ,L ;H ,F ) : Uε T∗(R S2) , (x) := ∈ × Y l l l l l → × Y ˆ(x) otherwise (cid:26) Y to a full neighbourhood of the zero section. In a final step the hamiltonian flow Φ : R P P generated the con- tinuous extension H : P R of the Hamilton×ian →function Hˆ : T∗Mˆ R → → is linearized by slightly changing the coordinates using the (real-analytic but incomplete) flow generated by Hˆ. See [Kn] forYdetails. 6 In summary, we obtain a real-analytic extension (2) of the Hamiltonian system (T∗Mˆ ,ωˆ,Hˆ ). Thus the Hamiltonian vector field X (defined by H i ω = dH) is real-analytic, too, and thus the flow Φ : R P P is known XH × → to be real-analytic (see, e.g. [Ho]), proving assertion 1. 2) This, however is insufficient to show real-analyticity of the Møller transformations and asymptotic momenta. It is known that even for smooth potentials these maps may be very non-smooth, see [Si]. There exists an energy-dependent virial radius R > 0, with vir d E ~q,~p > > 0 if ~q R (E) and E := H(p~,~q). (13) vir dt h i 2 k k ≥ If we assume q := ~q R (E) and ~q ,~p 0, then 0 0 vir 0 0 k k ≥ h i ≥ ~q(t) q t for all t 0, with λ := E/2/q (14) 0 λ 0 k k ≥ ·h i ≥ p and t := 1+(λt)2 , t := t . λ 1 h i h i h i In particular a trajectory lepaving the ball of radius R (E) cannot reenter vir this ball in the future but must go to spatial infinity. Analyticity estimates for the trajectory are then derived from the integral equation with initial conditions x = (p~ ,~q ) 0 0 0 t s ~q(t,x ) = ~q +tp~ V(~q(τ,x ))dτ ds, 0 0 0 0 − ∇ Z0 Z0 using the decay property of the potential |β|+1 C ∂βV(~q) β ! (β N3) ≤ | | ~q ∈ 0 (cid:18)k k(cid:19) (cid:12) (cid:12) (cid:12) (cid:12) outside the virial radius, and (14). Setting w~(t) w~ := sup k k λ k k t t≥0 λ h i we obtain for γ := (α,β), ∂γ := ∂α∂β x0 p0 q0 ∂γ ~q( ,x ) α!q−|β|+δ|γ|,1E−12|α|−1+δ|γ|,1. (15) k x0 · 0 kλ ≤ 0 7 This allows us to perform the (locally uniform w.r.t. x ) time limit in 0 ∂γ (p~(t,x ) p~ ) = (16) x0 0 − 0 − g t DN V(~q(τ,x )) ∂γ(1)~q(τ,x ),...,∂γ(N)~q(τ,x ) dτ. ∇ 0 x0 0 x0 0 NX=1γ(1)+.X..+γ(N)=γZ0 (cid:16) (cid:17) |γ(i)|>0 with g := γ 1, and to conclude that ~p+ is real-analytic at x . We can 0 | | ≥ substitute the assumption x s+ for the stronger assumptions q := q 0 0 0 ∈ k k ≥ R (E), ~q ,~p 0, as initial conditions meeting the first lead to data vir 0 0 h i ≥ meeting the second one after some t. The same holds for p~− by reversibility (p~−(p~ ,~q ) = p~+( p~ ,~q )). 0 0 0 0 − − The proof of real-analyticity of the Møller transforms is based on the integral equation ∞ ∞ Z ∂γ ~r(t) = ∂γ − ∞ V(~q(τ,x )) dτ ds x0 x0∇ ~ − 0  Zt Zs Q(τ;x0) (cid:13) (cid:13)  for~r(t;x ) := ~q(t,x ) Q~(t;x ) with(cid:13)Kepler h(cid:13)yperbola P~(t;x ),Q~(t;x ) = 0 0 0 (cid:13) (cid:13) 0 0 − Φt (X ). Inspecting the proof of smoothness for the(cid:16)Møller transform(cid:17)s in ∞ 0 [Kn], one sees that the estimates for ∂γ ~r can be dominated by γ !C|γ|. x0 | | 3) For a scattering state x s time delay equals ∈ τ(x) = 1 lim σ+ Φt(x) σ (R) Ω+ σ (R) Ω− (Φt(x))dt 2R→∞ R ◦ · ∞ ◦ ∗ − ∞ ◦ ∗ Z (cid:0) (cid:1) + 1 lim σ− Φt(x) σ (R) Ω− σ (R) Ω+ (Φt(x))dt, 2R→∞ R ◦ · ∞ ◦ ∗ − ∞ ◦ ∗ Z (cid:0) (cid:1) withσ±(p~,~q) := θ( ~q ~p),see[KK]andΩ± := (Ω±)−1. Usingtheintertwining ± · ∗ property Ω± Φt = Φt Ω± this can be written entirely in terms of Møller ∗ ◦ ∞ ◦ ∗ transformations and the Kepler flow: τ(x)=1 lim σ+ Φt (x) σ (R) Φt Ω+(x) σ (R) Φt Ω−(x) dt 2R→∞ R ◦ ∞ · ∞ ◦ ∞ ◦ ∗ − ∞ ◦ ∞ ◦ ∗ Z (cid:0) (cid:1) 1 lim σ− Φt (x) σ (R) Φt Ω+(x) σ (R) Φt Ω−(x) dt. −2R→∞ R ◦ ∞ · ∞ ◦ ∞ ◦ ∗ − ∞ ◦ ∞ ◦ ∗ Z (cid:0) (cid:1) With Assertion 2 and the analog of formula (10) this implies analyticity of τ. (cid:3) 8 4 Gevrey Integrals of Motion Wenowshowtheexistenceofindependent constantsofmotionforallenergies E > E . In [Kn] many estimates are shown to hold true above a threshold th energy E 0: th ≥ In [Kn] many estimates are shown to hold true above a threshold energy E 0: th ≥ For d = 2 and attracting centres (Z > 0) E = 0. k th • For d = 3, arbitrary Z = 0 and non-collinear configurations of the k • 6 ~s R3 the existence of such a threshold is proven. k ∈ In particular the set b of bounded orbits of energy E > E is shown to be E th of measure zero (and has a Cantor set structure for n 3). ≥ Accordingtothestandarddefinition(see, e.g.[AM],Def.5.2.20)functions f ,...,f : P˜ R on a symplectic manifold (P˜,ω˜) are called independent if 0 k the set of sing→ular points of F := f ... f : P˜ Rk has measure zero. 0 k × × → To simplify discussion, we set P˜ := H−1([E ,E ]) 1 2 for an arbitrary energy interval, E E E < , and fg f := H . th ≤ 1 ≤ 2 ∞ 0 ≡ 0 |P˜ Then for a parameter g > 1 we define for k = 1,2 p+(x)exp eC(g)hτ(x)i , x s fg : P˜ R , fg(x) := k − ∈ (17) k → k 0 , x s, (cid:26) (cid:0) (cid:1) 6∈ where C(g) = C2 with C to be defined below. Putting things together we g−1 2 get a map Fg : P˜ R3. → The notation is chosen so that Fg belongs to the Gevrey class of index g. Now we collect some information about Gevrey functions, which were introduced in [Ge]. Definition 1 For g 1 and an open set Ω Rn a function f C∞(Ω) is ≥ ⊂ ∈ called of Gevrey class g if for every compact K Ω there exist A ,C > 0 K K ⊂ with max ∂αf(x) A C|α|(α!)g (α Nn). x∈K| | ≤ K K ∈ 0 Here α = α +...+α and α! = α ! ... α ! if α = (α ,...,α ). 1 n 1 n 1 n | | · · The vector space of these functions is denoted by G (Ω). g 9 Then G (Ω) is the space of real-analytic functions, and G (Ω) G (Ω) 1 g′ g ⊃ for g′ > g. G (Ω) is stable w.r.t. partial derivatives, compositions, and the g implicit function theorem holds within the class. Forg > 1 Gevrey partitions of unity exist. Example 1 For g > 1 exp( x−1/(g−1)) , x > 0 fg(x) := − 0 , x 0, (cid:26) ≤ is a function in G (R) but not in G (R) for any g′ < g, see e.g. [Ju]. g g′ For a real-analytic manifold we can define the spaces of Gevrey functions, as the defining family of bounds is preserved by coordinate changes. Theorem 1 For all g > 1 the functions fg,fg,fg are independent, in invo- 0 1 2 lution and of Gevrey class g. Proof. In Thm. 12.8 of [Kn] it was shown that the set b of bounded orbits of E energy E is of Liouville measure zero for all E > E . As the set t := x th E { ∈ H−1(E) x T of trapped orbits is always of Liouville measure zero, on P˜ the set| P˜∈ s }is of measure zero. On P˜ s the functions fg and fg are \ ∩ 1 2 real-analytic, using Proposition 1, whereas they are zero on P˜ s. We study their decay near P˜ s in order to prove that they\are in Gg(P˜). ˜ \ All orbits in P s enter the interaction zone, so we need only orbits in P˜ s entering the \interaction zone. W.l.o.g. we assume R to be constant vir ∩ on [E ,E ]. Then the region 1 2 I˜:= x P˜ q(x) R vir { ∈ | k k ≤ } projecting to the interaction zone is compact. The restriction of the real- analytic flow Φ : R P R to a domain of the form [ ε,ε] I˜ thus has × → − × partial derivatives ∂αΦ(t,x) C˜ C˜|α|α!. We conclude, using Corollary 1 that for arbitrary t| R such|th≤at Φ1(t2,x) I˜, too, ∈ ∈ ∂αΦ(t,x) C exp(C α t ) α!. (18) 1 2 | | ≤ | | h i Next we analyse time delay (6) for orbits entering I˜. By compactness of I˜ there exists a R˜ R such that all Kepler orbits Φt Ω±(x) t R ≥ vir { ∞ ◦ ∗ | ∈ } 10