Scattering invariants in Euler's two-center problem
Nikolay Martynchuk, Holger R. Dullin, Konstantinos Efstathiou, Holger Waalkens
SSCATTERING INVARIANTS IN EULER’S TWO-CENTERPROBLEM
N. MARTYNCHUK , H. R. DULLIN , K. EFSTATHIOU , AND H. WAALKENS Abstract.
The problem of two fixed centers was introduced by Euleras early as in 1760. It plays an important role both in celestial me-chanics and in the microscopic world. In the present paper we studythe spatial problem in the case of arbitrary (both positive and negative)strengths of the centers. Combining techniques from scattering theoryand Liouville integrability, we show that this spatial problem has topo-logically non-trivial scattering dynamics, which we identify as scatteringmonodromy. The approach that we introduce in this paper applies moregenerally to scattering systems that are integrable in the Liouville sense.Keywords: Action-angle coordinates; Hamiltonian systems; Liouville in-tegrability; Scattering map; Scattering monodromy. Introduction
The problem of two fixed centers, also known as the Euler 3-body problem,is one of the most fundamental integrable problems of classical mechanics.It describes the motion of a point particle in Euclidean space under theinfluence of the Newtonian force field F = −∇ V, V = − µ r − µ r . Here r i are the distances of the particle to the two fixed centers and µ i arethe strengths (the masses or the charges) of these centers. We note that theKepler problem corresponds to the special cases when the centers coincideor when one of the strengths is zero.The (gravitational) Euler problem was first studied by L. Euler in a seriesof works in the 1760s [19–21]. He discovered that this problem is integrableby putting the equations of motion in a separated form. Elliptic coordinates,which separate the problem and which are now commonly used, appearedin his later paper [21] and, at about the same time, in the work of Lagrange[36]. The systematic use of elliptic coordinates in classical mechanics wasinitiated by Jacobi, who used a more general form of these coordinates tointegrate, among other systems, the geodesic flow on a triaxial ellipsoid; see[29] for more details. Johann Bernoulli Institute for Mathematics and Computer Science, University ofGroningen, P.O. Box 407, 9700 AK Groningen, The Netherlands. School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006,Australia. a r X i v : . [ m a t h - ph ] J a n N. MARTYNCHUK, H. R. DULLIN, K. EFSTATHIOU, AND H. WAALKENS
Since the early works of Euler and Lagrange the Euler problem and itsgeneralizations have been studied by many authors. First classically andthen, since the works of Pauli [45] and Niessen [43] in the early 1920s, alsoin the setting of quantum mechanics. We indicatively mention the works[5, 10, 14, 18, 47, 50–52]. For a historical overview we refer to [26, 44].In the present work we will be interested in the spatial Euler problem.For us, it will be important that this problem is a Hamiltonian system withtwo additional structures: it is a scattering system and it is also integrablein the Liouville sense . The structure of a scattering system comes from thefact that the potential V ( q ) → , (cid:107) q (cid:107) → ∞ , decays at infinity sufficiently fast (is of long range ). It allows one to comparea given set of initial conditions at t = −∞ with the outcomes at t = + ∞ .An introduction to the general theory of scattering systems can be foundin [11, 33]. Liouville integrability comes from the fact that the system is separable ; the three commuting integrals of motion are: • the energy function — the Hamiltonian, • the separation constant; see Subsection 2.1, • the component of the angular momentum about the axis connectingthe two centers.An introduction to the general theory of Liouville integrable systems can befound in in [3, 8, 33].Separately these two structures of the Euler problem have been discussedin the literature. Scattering has been studied, for instance, in [31, 47]. Thecorresponding Liouville fibration has been studied in [51] — from the per-spective of Fomenko theory [3,25], action coordinates and Hamiltonian mon-odromy [12]. We will consider both of the structures together and show thatthe Euler problem has non-trivial scattering invariants, which we will call purely scattering and mixed scattering monodromy , cf. [2, 13, 16, 32, 39]. Forcompleteness, the qualitatively different case of Hamiltonian monodromywill be also discussed. We note that the approach that we introduce in thepresent paper applies more generally to systems that are both scatteringand integrable in the Liouville sense.The paper is organized as follows. The problem is introduced in Sec-tion 2. Bifurcation diagrams are given in Section 3. In Section 4 we discussclassical potential scattering theory. In Section 5 we adapt the discussionof Section 4 to the context of scattering systems that are integrable in theLiouville sense. In particular, we give a definition of a reference system forintegrable systems. We note that the choice of a reference system is im-portant for the definition of scattering monodromy; see Subsection 5.2. Forthe Euler problem, scattering monodromy is discussed in detail in Section 6.Hamiltonian monodromy is addressed in Subsection 6.3. The main part ofthe paper is concluded with a discussion in Section 7. Additional details arepresented in the Appendix.
ULER’S TWO-CENTER PROBLEM 3 Preliminaries
We start with the 3-dimensional Euclidean space R and two distinctpoints in this space, denoted by o and o . Let q = ( x, y, z ) be Cartesiancoordinates in R and let p = ( p x , p y , p z ) be the conjugate momenta in T ∗ q R .The Euler two-center problem can be defined as a Hamiltonian system on T ∗ ( R \ { o , o } ) with a Hamiltonian function H given by(1) H = (cid:107) p (cid:107) V ( q ) , V ( q ) = − µ r − µ r , where r i : R → R is the distance to the center o i . The strengths of thecenters µ i can be both positive and negative; without loss of generality weassume that the center o is stronger, that is, | µ | ≤ | µ | . Remark 2.1.
When µ i > µ i <
0) the center o i is attractive (resp.,repulsive). The cases µ (cid:54) = µ = 0 and µ (cid:54) = µ = 0 correspond to a Keplerproblem. In the case µ = µ = 0 the dynamics is trivial and we have thefree motion ( t, q , p ) (cid:55)→ ( q + tp , p ).2.1. Separation and integrability.
Without loss of generality we assume o i = (0 , , ( − i a ) for some a >
0, so that, in particular, the fixed centers o and o are located on the z -axis in the configuration space. Rotationsaround the z -axis leave the potential function V invariant. It follows that(the z -component of) the angular momentum(2) L z = xp y − yp x commutes with H , that is, L z is a first integral. It is known [18, 52] thatthere exists another first integral given by(3) G = H + 12 ( L − a ( p x + p y )) + a ( z + a ) µ r − a ( z − a ) µ r , where L = L x + L y + L z is the squared angular momentum. The expressionfor the integral G can be obtained using separation in elliptic coordinates,as described below. It will follow from the separation procedure that thefunction G commutes both with H and with L z , which means that theproblem of two fixed centers is Liouville integrable .Consider prolate ellipsoidal coordinates ( ξ, η, ϕ ):(4) ξ = 12 a ( r + r ) , η = 12 a ( r − r ) , ϕ = Arg( x + iy ) . Here ξ ∈ [1 , ∞ ) , η ∈ [ − , ϕ ∈ R / π Z . Let p ξ , p η , p ϕ = L z be theconjugate momenta and l be the value of L z . In the new coordinates theHamiltonian H has the form(5) H = H ξ + H η ξ − η , where H ξ = 12 a ( ξ − p ξ + 12 a l ξ − − µ + µ a ξ N. MARTYNCHUK, H. R. DULLIN, K. EFSTATHIOU, AND H. WAALKENS and H η = 12 a (1 − η ) p η + 12 a l − η + µ − µ a η. Multiplying Eq. (5) by ξ − η and separating we get the first integral G = ξ H − H ξ = η H + H η . In original coordinates G has the form given in Eq. (3). Since L z = p ϕ , thefunction G commutes both with H and with L z .2.2. Regularization.
We note that in the case when one of the strengthsis attractive, collision orbits are present and, consequently, the flow of H on T ∗ ( R \ { o , o } ) is not complete. This complication is, however, notessential for our analysis since collision orbits, as in the Kepler case, canbe regularized. More specifically, there exists a 6-dimensional symplecticmanifold ( P, ω ) and a smooth Hamiltonian function ˜ H on P such that(1) ( T ∗ ( R \ { o , o } , dq ∧ dp ) is symplectically embedded in ( P, ω ),(2) H = ˜ H | T ∗ ( R \{ o ,o } ) ,(3) The flow of ˜ H on P is complete.This result is essentially due to [31, Proposition 2.3], where a similar state-ment is proved for the gravitational planar problem. The planar problemin the case of arbitrary strengths can be treated similarly (note that colli-sions with a repulsive center are not possible). The spatial case follows fromthe planar case since collisions occur only when L z = 0. We note that theintegrals L z and G can be also extended to P .One important property of the regularization is that the extensions of theintegrals to P , which will be also denoted by H , L z and G , form a completelyintegrable system . In particular, the Arnol’d-Liouville theorem [1] applies.In what follows we shall work on the regularized space P .3. Bifurcation diagrams
Before we move further and discuss scattering in the Euler problem, weshall compute the bifurcation diagrams of the integral map F = ( H, L z , G ),that is, the set of the critical values of this map. We distinguish two cases,depending on whether L z is zero or different from zero. The bifurcationdiagrams are obtained by superimposing the critical values found in thesetwo cases. By a choice of units we assume that a = 1.3.1. The case L z = 0 . Since L z = 0, the motion is planar. We assumethat it takes place in the xz -plane. Consider the elliptic coordinates ( λ, ν ) ∈ R × S [ − π, π ] defined by x = sinh λ cos ν, z = cosh λ sin ν. ULER’S TWO-CENTER PROBLEM 5
The level set of constant H = h, L z = l = 0 and G = g in these coordinatesis given by the equations p λ = 2 h cosh λ + 2( µ + µ ) cosh λ − g,p ν = − h sin ν − µ − µ ) sin ν + 2 g, where p λ and p ν are the momenta conjugate to λ and ν. The value ( h, , g )is critical when the Jacobian matrix corresponding to these equations doesnot have a full rank. Computation yields lines (cid:96) = { g = h + µ − µ , l = 0 } , (cid:96) = { g = h + µ − µ , l = 0 } and (cid:96) = { g = h + µ, l = 0 } , µ = µ + µ , and two curves { g = µ cosh λ/ , h = − µ/ λ, l = 0 } , { g = ( µ − µ ) sin ν/ , h = ( µ − µ ) / ν, l = 0 } . Points that do not correspond to any physical motion must be removedfrom the obtained set (allowed motion corresponds to the regions where thesquared momenta are positive).
Remark 3.1.
The corresponding diagrams in the planar problem are givenin Appendix B; see Fig. 5 and 6. We note that in the planar case the set ofthe regular values of F consists of contractible components and hence thetopology of the regular part of the Liouville fibration is trivial. Interestingly,this is not the case if the dimension of the configuration space is n = 3.We note that the singular Liouville foliation has non-trivial topology al-ready in the planar case. The corresponding bifurcations, in the sense ofFomenko theory [3, 4, 23–25], have been studied in [30, 51].3.2. The case L z (cid:54) = 0 . In order to compute the critical values in this caseit is convenient to use the ellipsoidal coordinates ( ξ, η ). (We note that for L z (cid:54) = 0 the z -axis is inaccessible, so ( ξ, η ) are non-singular.) The level setof constant H = h, L z = l and G = g in these coordinates is given by theequations p ξ = ( ξ − hξ + 2( µ + µ ) ξ − g ) − l ( ξ − ,p η = (1 − η )( − hη − µ − µ ) η + 2 g ) − l (1 − η ) . The value ( h, l, g ) with l (cid:54) = 0 is critical when the corresponding Jacobianmatrix does not have a full rank. Computation yields the following sets ofcritical values: (cid:26) g = h (2 ξ −
1) + ( µ + µ )(3 ξ − ξ , l = − ( µ + µ + 2 hξ )( − ξ ) ξ (cid:27) , (cid:26) g = h (2 η −
1) + ( µ − µ )(3 η − η , l = − ( µ − µ + 2 hη )( − η ) η (cid:27) , N. MARTYNCHUK, H. R. DULLIN, K. EFSTATHIOU, AND H. WAALKENS
Figure 1.
Positive energy slices of the bifurcation diagramfor the spatial Euler problem, attractive case. The blackpoints correspond to the critical lines (cid:96) i .where ξ > − < η <
1. As above, points that do not correspond toany physical motion must be removed.Representative positive energy slices in the gravitational case 0 < µ < µ are given in Fig. 1. The case of arbitrary strengths µ i is similar. Thestructure of the corresponding diagrams can partially be deduced from thediagrams obtained in the planar case; see Appendix B.4. Classical scattering theory
In this section we discuss certain qualitative aspects of scattering theoryfollowing [32, 33]. In Section 5 we explain how the theory can be adapted tothe context of scattering systems that are integrable in the Liouville sense,with the Euler problem as the leading example.4.1.
Preliminary remarks.
Classical scattering theory goes back to theworks of Cook [6], Hunziker [28] and Simon [48]. Since then it has receivedconsiderable interest and has been actively developed in several directions;see [2, 11, 13, 27, 32].In the framework of classical scattering one considers two Hamiltonianfunctions H and H r such that their flows become similar ‘at infinity’. Thisallows one can compare a given distribution of particles, that is, initialconditions, at t = −∞ with their final distribution at t = + ∞ . To be morespecific, consider a pair of Hamiltonians on T ∗ R n given by H = 12 (cid:107) p (cid:107) + V ( q ) and H r = 12 (cid:107) p (cid:107) + V r ( q ) , ULER’S TWO-CENTER PROBLEM 7 where the (singular) potentials V and V r are assumed to satisfy a certaindecay assumption; see Subsection 4.2. For scattering Hamiltonians the com-parison will be achieved in two steps. First we shall parametrize the pos-sible initial and final distributions using the flow of the ‘free’ Hamiltonian H = (cid:107) p (cid:107) . Then, for a given invariant manifold, we shall construct the scattering map , where only H and H r are compared. Remark 4.1.
One reason for such a procedure is the following. As we shallsee later in Section 5 and Appendix C, the ‘free’ Hamiltonian is not a naturalreference Hamiltonian for the Euler problem, unless the strengths µ = µ .However, the ‘free’ Hamiltonian will be convenient for the definition of theasymptotic states. Remark 4.2.
In what follows we sometimes refer to
H, H r as scatteringHamiltonians and to H r is a reference Hamiltonian for H . We note that the‘reference’ dynamics of H r is usually chosen to be simpler than the ‘original’dynamics of H .4.2. Decay assumptions.
In classical potential scattering the potentialfunction V : R n → R of a Hamiltonian H = (cid:107) p (cid:107) + V ( q ) is assumed todecay according to one of the following estimates:1. Finite-range: supp( V ) ⊂ R n is compact;2. Short-range case: | ∂ k V ( q ) | < c ( (cid:107) q (cid:107) + 1) −| k |− ε ;3. Long-range case: | ∂ k V ( q ) (cid:107) < c ( (cid:107) q (cid:107) + 1) −| k |− − ε . Here c and ε are positive constants, k = ( k , . . . , k n ) ∈ N n is a multi-index, | k | = k + . . . + k n is a norm of k and (cid:107) q (cid:107) denotes the Euclidean norm of q .For instance, any Kepler potential is of long range and the same is true ofthe potential found in the Euler problem.We will assume the potentials V and V r are short-range relative to somedecaying rotationally symmetric potentials (cid:101) V and (cid:101) V r , respectively. For thepotential V this means that (cid:107) ∂ k V − ∂ k (cid:101) V (cid:107) < c (cid:107) q (cid:107) −| k |− − ε , where (cid:101) V is rotationally symmetric with (cid:101) V ( q ) → , (cid:107) q (cid:107) → ∞ . A similarestimate is assumed to hold for V r − (cid:101) V r . Remark 4.3.
The potentials (cid:101) V and (cid:101) V r are needed to guarantee that theasymptotic direction and the impact parameter are defined and parametrizethe scattering trajectories in a continuous way. This is known to be the casefor short-range potentials V [33]. Our case reduces to the case of symmetricpotentials and in that case the statement follows from the conservation ofthe angular momentum.4.3. Asymptotic states.
The Hamiltonian flow g tH : P → P of H parti-tions the (regularized) phase space P into the following invariant subsets: b ± = { x ∈ P | sup t ∈ R ± (cid:107) g tH ( x ) (cid:107) < ∞} and s ± = { x ∈ P | H ( x ) > } \ b ± . N. MARTYNCHUK, H. R. DULLIN, K. EFSTATHIOU, AND H. WAALKENS
The invariant subsets b = b + ∩ b − , s = s + ∩ s − and trp = ( b + \ b − ) ∪ ( b − \ b + )are the sets of the bound , the scattering and the trapped states, respectively.We note that s − , s + and hence s = s − ∩ s + are open subsets of P .If the potential V is short-range relative to a decaying rotationally sym-metric potential, then the following limitsˆ p ± ( x ) = lim t →±∞ p ( t, x ) and q ±⊥ ( x ) = lim t →±∞ (cid:18) q ( t, x ) − (cid:104) q ( t, x ) , ˆ p ± ( x ) (cid:105) ˆ p ± ( x )2 h (cid:19) , where h = H ( x ) > g tH ( x ), are defined for any x ∈ s ± anddepend continuously on x . These limits are called the asymptotic direction and the impact parameter of the trajectory g tH ( x ), respectively. We note thatan asymptotic direction is always orthogonal to the corresponding impactparameter. Due to the g tH -invariance of ˆ p ± and q ±⊥ , we have the maps A ± = (ˆ p ± , q ±⊥ ) : s/g tH → AS from s/g tH to the asymptotic states AS ⊂ R n × R n . Here s/g tH is the space oftrajectories in s , that is, it is a quotient space of s by the equivalence relationwhere two points are considered equivalent if and only if they belong to asingle trajectory g tH ( x ). Similarly, one can construct the maps A ± r = (ˆ p ± , q ±⊥ ) : s r /g tH r → AS for the ‘reference’ Hamiltonian H r = p + V r ( q ) . Scattering map.
Using the maps A ± and A ± r constructed in Sub-section 4.3, we can now define the notion of a scattering map for a giveninvariant submanifold R of s . Definition 4.4.
Let R be a g tH -invariant submanifold of s and B = R/g tH .Assume that the composition map S = ( A − ) − ◦ A − r ◦ ( A + r ) − ◦ A + is well defined and maps B to itself. The map S is called the scattering map (w.r.t. H, H r and B ). Remark 4.5.
Due to the decay assumptions the maps A ± : s/g tH → AS and A ± r : s r /g tH → AS are homeomorphisms onto their images in AS . It follows that the scatteringmap S : B → B is a homeomorphism as well. Here the sets s/g tH , s/g tH and B are endowed with the quotient topology. ULER’S TWO-CENTER PROBLEM 9
Knauf ’s topological degree.
To get qualitative information aboutthe scattering it is useful to look at topological invariants of the scatteringmap. An important example in the context of general scattering theory is
Knauf ’s topological index ; see [32, 34]. We shall now recall its definition.Consider the case when the potential V is short-range relative to V r = 0.Let h > non-trapping energy , that is, a positive energy such that theenergy level H − ( h ) contains no trapping states, and let R = H − ( h ) ∩ s bethe intersection of the level H − ( h ) with the set s of the scattering states.There is the following result. Theorem 4.6. ([11,32])
The scattering manifold B = R/g tH is the cotangentbundle T ∗ S n − , where S n − is the sphere of asymptotic directions. Thecorresponding scattering map S h : B → B is a symplectic transformation of T ∗ S n − . Knauf’s topological degree is defined as a topological invariant of S h .Specifically, let Pr : T ∗ S n − → S n − be the canonical projection and S n − p = T ∗ p S n − ∪ {∞} be the one-point compactification of the cotangent space T ∗ p S n − . Definition 4.7. (Knauf, [32]) The degree deg( h ) of the energy h scatteringmap S h is defined as the topological degree of the mapPr ◦ S h : S n − p → S n − . Remark 4.8.
We note that by continuity deg( h ) is independent of thechoice of the initial direction p ∈ S n − ; see [32].The following theorem shows that for regular (that is, everywhere smooth)potentials deg( h ) is either 0 or 1, depending on the value of the energy h ; seeFig. 2. We note that for singular potentials, such as the Kepler potential,values different from 0 and 1 may appear. Theorem 4.9. (Knauf-Krapf, [34])
Let V be a regular short-range potentialand h > be a non-trapping energy. Then deg( h ) = (cid:40) , h ∈ (sup V, ∞ ) , , h ∈ (0 , sup V ) . Remark 4.10.
For the Euler problem with µ µ (cid:54) = 0, Knauf’s degree is notdefined (every positive energy h is trapping). Moreover, the free flow is nota proper reference unless µ = µ ; see Section 5. Nonetheless, as we shallshow in Sections 5 and 6, for a proper choice of a reference Hamiltonian anda scattering manifold, an analogue of Knauf’s degree can be defined. Figure 2.
Scattering at different energies. At high energiesdeg( h ) = 0 (left), at low energies deg( h ) = 1 (right).5. Scattering in integrable systems
The goal of the present section is to recast the above theory of scatteringin the context of Liouville integrability. The approach developed in thepresent section will be applied to the Euler problem in Section 6.5.1.
Reference systems.
As we have seen in Section 4, reference systemscan be used to define a scattering map, which is a map between the as-ymptotic states at t = −∞ and t = + ∞ of a given invariant manifold.For integrable systems, natural invariant manifolds are the fibers of the cor-responding integral map F and various unions of these fibers. It is thusnatural to require that the flow of a reference Hamiltonian maps the set ofasymptotic states of a given fiber of F to the set of asymptotic states of thesame fiber. This leads to the following definition. Definition 5.1.
Consider a scattering Hamiltonian H which gives rise to anintegrable system F : P → R n . A scattering Hamiltonian H r will be calleda reference Hamiltonian for this system if F (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = F (cid:18) lim t →−∞ g tH r ( x ) (cid:19) for every scattering trajectory t (cid:55)→ g tH r ( x ). Remark 5.2.
We note that Definition 5.1 can be generalized to the case ofscattering and integrable systems defined on abstract symplectic manifolds.However, for the purpose of the present paper it is sufficient to assume that H and H r are as in Section 4. Remark 5.3.
In scattering theory it is usually required that the flow of areference Hamiltonian maps the set of asymptotic states of a given energylevel to itself, which is a less restrictive assumption. Our point of view is thatfor integrable systems conserved quantities, such as the angular momentum,should also be taken into account.
ULER’S TWO-CENTER PROBLEM 11
A series of examples of reference Hamiltonians in the above sense is givenby rotationally symmetric potentials. This follows from the conservation ofangular momentum. Another example is the Euler problem. We recall thatthe Hamiltonian of this problem is given by H = (cid:107) p (cid:107) − µ r − µ r . Let F = ( H, L z , G ) : P → R be the integral map defined in Section 2. Wehave the following result. Theorem 5.4.
Among all Kepler Hamiltonians only H r = 12 p − µ − µ r and H r = 12 p − µ − µ r are reference Hamiltonians of the Euler problem F = ( H, L z , G ) . In partic-ular, the free Hamiltonian is a reference Hamiltonian of the Euler problemonly in the case µ = µ .Proof. See Appendix C. (cid:3)
Remark 5.5.
It follows from Theorem 5.4 that a Kepler Hamiltonian withthe strength µ + µ is not a reference of F = ( H, L z , G ), no matter where thecenter of attraction, resp., repulsion, is located. For the strength µ + µ andonly for this strength, the difference between the potentials is short-range.This implies that the Møller transformations (or the wave transformations )[11,33] are not defined with respect to the reference Hamiltonians H r i , unlessthe reference flow is appropriately modified. We note that the existence ofMøller transformations is important for the study of quantum scattering inthis problem.5.2. Scattering invariants.
Consider the Liouville fibration F : s → R n .Let H r be a reference Hamiltonian for F such that A ± ( s ) ⊂ A ± ( s r ) holds.Setting R = s , we get the scattering map S : B → B, B = R/g tH . The scattering map S allows to identify the asymptotic states of s at t = + ∞ with the asymptotic states at t = −∞ . This results in a new total space s c .We observe that under this identification the asymptotic states of a givenfiber of F : s → R n are mapped to the asymptotic states of the same fiber.This implies that s c is naturally fibered by F . The resulting fibration willbe denoted by F c : s c → R n . We note that the invariants of the fibration F c contain essential informationabout the scattering dynamics. One such invariant is scattering monodromy which we define as follows. Definition 5.6.
Assume that F c : s c → R n is a torus bundle. The (usual) monodromy of this torus bundle will be called scattering monodromy of the fibration F . Remark 5.7.
We note that scattering monodromy in the above sense isrelated to non-compact monodromy introduced in [16] for unbound systemswith focus-focus singularities. It is known that focus-focus singularities comewith a circle action [54]. One can use this (global) action to compactify thefibration F near a focus-focus fiber.5.3. Planar potential scattering.
Here we shall discuss the case n = 2of planar scattering systems. The goal is to relate our notion of scatteringmonodromy to the existing definition in terms of the deflection angle [2, 13]and to make an explicit connection to the scattering map.Assume that V and V r are rotationally symmetric, that is, V ( q ) = W ( (cid:107) q (cid:107) ) and V r ( q ) = W r ( (cid:107) q (cid:107) ) for some W, W r : R + → R . Then the angular momentum L z = xp y − yp x is conserved. Let F = ( H, L z )be the integral map of the original system and N be an arbitrary submanifoldof the non-trapping set(6) N T = { ( h, l ) ∈ image( F ) | F − ( h, l ) ⊂ s } . The manifold F − ( N ) is an invariant submanifold of the phase space P ,which contains no trapping states (it consist of scattering states only).Consider the case when N = γ is a regular simple closed curve in N T .Let R = F − ( γ ) and S : B → B, B = F − ( γ ) /g tH , denote the correspondingscattering map. Then we have the following result. Theorem 5.8.
The following statements are equivalent.(1) The scattering monodromy along γ is a Dehn twist of index m ;(2) The variation of the deflection angle along γ equals πm ;(3) The scattering map S is a Dehn twist of index m . Remark 5.9.
By a Dehn twist of index m we mean a homeomorphism of a2-torus such that its push-forward map is given by (the conjugacy class of)the matrix M = (cid:18) m (cid:19) . We note that the scattering manifold B is a 2-torus in this case. Remark 5.10.
The total deflection angle of a trajectory g tH ( x ) = ( q ( t ) , p ( t ))is defined by Φ = + ∞ (cid:90) −∞ dϕ ( q ( t )) dt dt, where ϕ is the polar angle in the configuration xy -plane. The deflection angle is defined as the difference of the total deflection angles for the original andthe reference trajectories. We note that (2) is essentially the definition ofscattering monodromy due to [2, 13]. ULER’S TWO-CENTER PROBLEM 13
Proof. (1) ⇔ (2) . Let ( a, b ) be homology cycles on the fiber F − c ( γ ( t )) suchthat b corresponds to the circle action given by L z . Transporting the cyclesalong γ we get b (cid:55)→ b and a (cid:55)→ a + mb for some integer m . But the differenceΦ − Φ r = + ∞ (cid:90) −∞ dϕ ( q ( t )) dt dt − + ∞ (cid:90) −∞ dϕ ( q r ( t )) dt dt, where g tH r = ( q r ( t ) , p r ( t )) is a reference trajectory with the same energy andangular momentum, can be seen as the rotation number on the fibers of F c .It follows that the variation of Φ − Φ r along γ equals 2 πm .(2) ⇔ (3) . The scattering map S allows one to consider the compactifiedtorus bundle Pr : F − ( γ ) c → S = R ∪ {∞} , where R corresponds to the time. The torus bundle considered in (1) hasthe same total space, but is fibered over γ . Suppose that the monodromy ofthis bundle is given by the matrix M = (cid:18) m (cid:19) . Then the monodromy of Pr : F − ( γ ) c → S is the same, for otherwise thetotal spaces would be different. The result follows. (cid:3) Remark 5.11.
We note that in the original definition of [13] the potential V is assumed to be repulsive and V r = 0 . In this case the equivalence (1) ⇔ (2)follows from the results of [16].Theorem 5.8 gives three alternative definitions of monodromy in the caseof scattering integrable systems in the plane. We observe that for the originaldefinition in terms of the deflection angle (Definition (2)) it is important thatthe scattering takes plane in the plane. On the other hand, from Section 4and the present section it follows that Definitions (1) and (3) are suitablefor scattering integrable systems with many degrees of freedom, such as theEuler problem. Definition (3), similarly to Knauf’s degree, can be naturallyapplied to scattering systems even without integrability.6. Scattering in the Euler problem
In this section we study scattering in the Euler problem using the refer-ence Kepler Hamiltonians identified in the previous section. We will showthat the Euler problem has non-trivial scattering monodromy of two dif-ferent kinds: purely scattering monodromy and another kind, where bothscattering and Hamiltonian monodromy are non-trivial. The latter kind canbe observed only if the number of degrees of freedom n ≥
3. Purely Hamil-tonian monodromy is also present in the problem; it survives the limitingcases of vanishing µ i , including the free flow. Scattering monodromy (ofboth kinds) is trivial for the free flow. However, scattering monodromy ofthe second kind is still present in the Kepler problem. Scattering map.
Let F = ( H, L z , G ) denote the integral map of theEuler problem. Let N be a submanifold of(7) N T = { ( h, l, g ) ∈ image( F ) | F − ( h, l, g ) ⊂ s } . The manifold F − ( N ) is an invariant submanifold of the phase space P ,which contains scattering states only. Following the construction in Sections4 and 5, we can define the scattering maps S : B → B with respect to H ,the reference Kepler Hamiltonian H r = H r or H r = H r , where H r = 12 p − µ − µ r and H r = 12 p − µ − µ r , and B = F − ( N ) /g tH as in Subsection 4.4. Remark 6.1.
We recall that the scattering map S is defined by S = ( A − ) − ◦ A − r ◦ ( A + r ) − ◦ A + , where A ± = (ˆ p ± , q ±⊥ ) : s ± /g tH → AS and A ± r = (ˆ p ± , q ±⊥ ) : s ± r /g tH → AS map s ± ⊂ P and s ± r to the asymptotic states AS . Here the index r refersto a reference system ( H r or H r in our case). Remark 6.2.
We note that the potential V = − µ r − µ r of the Euler problem is short-range relative to (cid:101) V ( q ) = − ( µ + µ ) / (cid:107) q (cid:107) , whichis a Kepler potential. The reference potentials are Kepler potentials and aretherefore rotationally symmetric. It follows that the decay assumptions ofSubsection 4.2 are met.6.2. Scattering monodromy.
First we consider the case of a gravitationalproblem (0 < µ < µ ) with H r = H r as the reference Kepler Hamiltonian.The other cases can be treated similarly; see Subsection 6.4.For sufficiently large h the h = h slice of the bifurcation diagram hasthe form shown in Fig. 3. Let γ i , i = 1 , , , be a simple closed curve in N T h = { ( h, g, l ) ∈ N T | h = h } that encircles the critical line (cid:96) i , where (cid:96) = { g = h + ( µ − µ ) , l = 0 } , (cid:96) = { g = h + ( µ − µ ) , l = 0 } and (cid:96) = { g = h + ( µ + µ ) , l = 0 } . For each γ i , consider the torus bundle F i : E i → γ i , where the total space E i is obtained by gluing the ends of the fibers of F over γ i via the scatteringmap S . We recall that scattering monodromy along γ i with respect to H r is defined as the usual monodromy of the torus bundle F i : E i → γ i ; seeDefinition 5.6 and Appendix A. ULER’S TWO-CENTER PROBLEM 15
Figure 3.
Energy slice of the bifurcation diagram for thespatial Euler problem, attractive case.
Remark 6.3.
Alternatively, one can define F i : E i → γ i by gluing the fibersof the original and the reference integral maps at infinity. Both definitionsare equivalent in the sense that the monodromy of the resulting torus bundlesare the same.Consider a starting point γ i ( t ) ∈ γ i in the region where l >
0. Wechoose a basis ( c ξ , c η , c ϕ ) of the first homology group H ( F − i ( γ i ( t ))) (cid:39) Z as follows. The cycle c ξ = c oξ ∪ c rξ is obtained by gluing the non-compact ξ -coordinate lines c oξ for the original and c rξ for the reference systems atinfinity. In other words, for we glue the lines p ξ = ( ξ − hξ + 2( µ + µ ) ξ − g ) − l ( ξ − on F − ( γ i ( t )) , γ i ( t ) = ( h, g, l ), and p ξ = ( ξ − hξ + 2( µ − µ ) ξ − g ) − l ( ξ − on the reference fiber F − r ( γ i ( t )) at the limit points ξ = ∞ , p ξ = ±√ h. The cycles c η and c ϕ are such that their projections onto the configurationspace coincide with coordinate lines of η and ϕ , respectively. In other words, the cycle c η on F − ( γ i ( t )) is given by p η = (1 − η )( − hη − µ − µ ) η − g ) − l (1 − η ) and c ϕ is an orbit of the circle action given by the Hamiltonian flow of themomentum L z . We have the following result. Theorem 6.4.
The monodromy matrices M i of E i → γ i with respect to thenatural basis ( c ξ , c η , c ϕ ) have the form M = , M = −
10 1 10 0 1 and M = . Proof.
Case 1 , loop γ . First we note that the cycle c ϕ is preserved underthe parallel transport along γ . This follows from the fact that L z generatesa free fiber-preserving circle action on E i . The cycles c ξ and c η can benaturally transported only in the regions where l (cid:54) = 0. We thus need tounderstand what happens at the critical plane l = 0.Let R > E ,R = { x ∈ E | ξ ( x ) > R } has exactly two connected components, which we denote by E +1 ,R and E − ,R .We define a 1-form α on (a part of) E i by the formula α = pdq − χ ( ξ ) p ξ ( h, g, l, ξ ) dξ, where χ ( ξ ) is a bump function such that(i) χ ( ξ ) = 0 when ξ < R ;(ii) χ ( ξ ) = 1 when ξ > R. The square root function p ξ ( h, g, l, ξ ) is assumed to be positive on E +1 ,R andnegative on E − ,R . By construction, the 1-form α is well-defined and smoothon E i outside collision points. Since dα = dp ∧ dq = − ω on F − ( γ i ) ∪ F − r ( γ i ) ⊂ E i , we have that dα = 0 on each fiber of F i .Consider the modified actions with respect to the form α : I ϕ = 12 π (cid:90) c ϕ α, I η = 12 π (cid:90) c η α and I modξ = 12 π (cid:90) c ξ α. The modified actions are well defined and, in view of dα = 0, depend onlyon the homology classes of c ξ , c η and c ϕ . It follows that I ϕ and I η coincidewith the ‘natural’ actions (defined as the integrals over the usual 1-form pdq ). We note that the ‘natural’ ξ -action I ξ = 12 π (cid:90) c ξ pdq ULER’S TWO-CENTER PROBLEM 17 diverges, cf. [13]. From the continuity of the modified actions at l = 0 itfollows that the corresponding scattering monodromy matrix has the form M = m m . Since the modified actions do not have to be smooth at l = 0, the integers m and m are not necessarily zero. In order to compute these integers weneed to compare the derivatives ∂ l I η and ∂ l I ξ at l → ± . A computation ofthe corresponding residues giveslim l →± ∂ l I η = lim l →± π ∂ l (cid:90) c η pdq = (cid:40) , when g < h + µ − µ , ∓ / , when µ − µ < g − h < µ − µ , andlim l →± ∂ l I modξ = lim l →± π ∂ l (cid:90) c oξ pdq − π ∂ l (cid:90) c rξ pdq − lim l →± π (cid:90) c ξ χ ( ξ ) p ξ ( h, g, l, ξ ) dξ = 0(for the two ranges of g ). It follows that m = 0 and m = 1. Case 2 , loop γ . This case is similar to Case 1 . The correspondinglimits are given bylim l →± ( ∂ l I η , ∂ l I modξ ) = (cid:40) ( ∓ / , , when µ − µ < g − h < µ − µ , ( ∓ , ± / , when µ − µ < g − h < µ + µ . Case 3 , loop γ . The computation in this case is also similar to Case 1 .The corresponding limits are given bylim l →± ( ∂ l I η , ∂ l I modξ ) = (cid:40) ( ∓ , ± / , when µ − µ < g − h < µ + µ , ( ∓ , , when h + µ + µ < g. (cid:3) Remark 6.5.
One difference between
Case 3 and the other cases is thetopology of the critical fiber, around which scattering monodromy is defined.In
Case 3 the critical fiber is the product of a pinched cylinder and a circle,whereas in the other cases it is the product of a pinched torus and a real line.This implies, in fact, that
Case 3 is purely scattering, whereas in the othercases Hamiltonian monodromy is present; see Subsection 6.3 for details.
Remark 6.6.
Theorem 6.4 admits the following geometric proof in thepurely scattering case.
Proof for Case 3 of Theorem 6.4.
The action I (cid:48) η = (cid:40) I η , if l ≥ I η − l, if l < . is smooth and globally defined (over γ ). Moreover, the corresponding circleaction extends to a free action in F − ( D ), where D ⊂ N T h is a 2-disksuch that ∂D = γ . Since there is also a circle action given by I ϕ , the resultcan be also deduced from the general theory developed in [17, 38]. (cid:3) We note that from the last proof it follows that the choice of a referenceKepler Hamiltonian does not affect the result in the purely scattering case.This agrees with the point of view presented recently in [16] for two degreeof freedom systems with focus-focus singularities. For the curves γ and γ ,it is important which of the two reference Kepler Hamiltonians is chosen;see Subsection 6.4.As a corollary, we get the following result for the scattering map in thepurely scattering case of the curve γ . Theorem 6.7.
The scattering map S : B → B , where B = F − ( γ ) /g tH , is a Dehn twist. The push-forward map is conjugate in SL (3 , Z ) to S (cid:63) = . Proof.
The proof is similar to the proof of the equivalence (2) ⇔ (3) given inTheorem 5.8. The scattering map S allows one to consider the compactifiedtorus bundle Pr : F − ( γ ) c → S = R ∪ {∞} , where R corresponds to the time. The torus bundle F : E → γ has thesame total space, but is fibered over γ . By Theorem 6.4, the monodromyof the bundle F : E → γ is given by the matrix M = . Then the monodromy of the first bundle Pr : F − ( γ ) c → S is the same,for otherwise the total spaces would be different. The result follows. (cid:3) Remark 6.8.
It follows from the proof and Subsection 6.4 that Theorem 6.7holds for any µ i (cid:54) = 0 and for any regular closed curve γ ⊂ N T such that1. The energy value h is positive on γ ;2. γ encircles the critical line { g = h + µ + µ , l = 0 } exactly onceand does not encircle any other line of critical values;3. γ does not cross critical values of F . ULER’S TWO-CENTER PROBLEM 19
Figure 4.
Energy slice of the bifurcation diagram for thespatial Euler problem, attractive case.It can be shown that such a curve γ always exists; an example is givenin Fig. 4. We note that the third condition can be weakened in the case − µ < µ <
0. In this case the attraction of µ dominates the repulsion of µ and, as a result, bound motion coexists with unbound motion for a rangeof positive energies. Instead of F − ( γ ) one may consider its unboundedcomponent.6.3. Topology.
As we have noted before, alongside scattering monodromy,the Euler problem admits also another type of invariant — Hamiltonianmonodromy. Here we consider the generic case of | µ | (cid:54) = | µ | (cid:54) = 0 in the caseof positive energies. The case of negative energies is similar — it has beendiscussed in detail in [51]. The critical cases can be easily computed fromthe generic case by considering curves that encircle more than one of thesingular lines (cid:96) = { g = h + ( µ − µ ) , l = 0 } , (cid:96) = { g = h + ( µ − µ ) , l = 0 } and (cid:96) = { g = h + ( µ + µ ) , l = 0 } . Let γ i be a closed curve that encircles only the critical line (cid:96) i ; see Fig. 4. Thefibration F : F − ( γ i ) → γ i is a T × R -bundle. The following theorem showsthat the Hamiltonian monodromy (see Appendix A) is non-trivial along thecurves γ and γ and is trivial along γ . Theorem 6.9.
The Hamiltonian monodromy of F : F − ( γ i ) → γ i , i = 1 , ,is conjugate in SL (2 , Z ) ⊂ SL (3 , Z ) to M = . Here the right-bottom × block acts on T and the left-top × block actson R .Proof. The result follows from the proof of Theorem 6.4. For completeness,we give an independent proof below.After the reduction of the surface H − ( h ) with respect to the flow g tH weget a singular T torus fibration over a disk D i , ∂D i = γ i , with exactly onefocus-focus point. The result then follows from [37, 41, 54]. This argumentapplies to both of the lines (cid:96) and (cid:96) . Since the flow of L z gives a global circleaction, the monodromy matrix M is the same in both cases; see [9]. (cid:3) Theorem 6.10.
The Hamiltonian monodromy of F : F − ( γ ) → γ is triv-ial.Proof. Observe that the Hamiltonian flows of I ϕ ,I (cid:48) η = (cid:40) I η , if l ≥ I η − l, if l < . and H generate a global T × R action on F − ( γ ). It follows that the bundle F : F − ( γ ) → γ is principal. Since γ is a circle, it is also trivial. (cid:3) We note that Hamiltonian monodromy is an intrinsic invariant of theEuler problem, related to the non-trivial topology of the integral map F .Interestingly, it is also present in the critical cases:(1) µ = µ (symmetric Euler problem) [51],(2) µ or µ = 0 (Kepler problem) [15] and(3) µ = µ = 0 (the free flow).In the case of bound motion (1) and (2) are due to [51] and [15], respectively.From the scattering perspective Hamiltonian monodromy is recovered if oneconsiders the original Hamiltonian H also as a reference.6.4. General case.
Here we consider the case of of arbitrary strengths µ i .We observe that the scattering monodromy matrices with respect to thereference Kepler Hamiltonians H r and H r are necessarily of the form m n for some integers m and n . These integers (for different choices of thestrengths µ i and the critical lines (cid:96) i ) are given in Table 1. ULER’S TWO-CENTER PROBLEM 21 γ γ γ Scattering monodromy w.r.t. H r Generic | µ | (cid:54) = | µ | (cid:54) = 0 m = − , n = 1 m = 0 , n = 1 m = 1 , n = 0 Critical − µ = µ < m = − , n = 1 m = 0 , n = 1 n = 1 , n = 00 < µ = µ m = − , n = 2 m = 1 , n = 0 µ = µ < m = − , n = 2 m = 1 , n = 0 µ = µ = 0 m = 0 , n = 20 = µ < µ n = 1 m = 0 , n = 1 m = 0 ,µ < µ = 0 m = − , n = 1 m = 1 , n = 1 Scattering monodromy w.r.t. H r Generic | µ | (cid:54) = | µ | (cid:54) = 0 m = 0 , n = 1 m = − , n = 1 m = 1 , n = 0 Critical − µ = µ < m = 0 , n = 1 m = − , n = 1 m = 1 , n = 00 = µ < µ n = 1 m = − , n = 1 m = 1 ,µ < µ = 0 m = 0 , n = 1 m = 0 , n = 1 Table 1.
Scattering monodromy, general case.
Remark 6.11.
We note that one can compute the monodromy matrices inthe critical cases from the matrices found in the generic cases. Specifically,it is sufficient to consider the curves that encircle more than one critical line (cid:96) i and multiply the monodromy matrices found around each of these lines.For instance, the monodromy matrix around the curve g = h in the free flowequals the product of the three monodromy matrices found in (any) genericEuler problem. 7. Discussion
In the present paper we have shown that the spatial Euler problem,alongside non-trivial Hamiltonian monodromy [51], has non-trivial scatter-ing monodromy of two different types: pure and mixed scattering mon-odromy. The first type reflects the presence of a special periodic orbit — acollision orbit that bounces between the two centers — and the associatedtrapping trajectories. In the spatial case one can go around these trajec-tories and compare the flow at infinity to an appropriately chosen Keplerproblem. Scattering monodromy of the second type is related to the differ-ence in dynamics of the original and the reference systems; here in additionto scattering monodromy also Hamiltonian monodromy is present. Interest-ingly, scattering monodromy of the second type survives vanishing of one ofthe centers: it can be also observed in the limiting case of attractive and repulsive Kepler problems H r = 12 p − µr and H r = 12 p + µr . Hamiltonian monodromy is present not only in the Kepler problem [15], butalso in the free flow. The purely scattering monodromy is special to thegenuine Euler problem; we conjecture that this invariant is also present inthe restricted three-body problem.8.
Acknowledgements
We would like to thank Prof. Dr. A. Knauf for the useful and stimulatingdiscussions.
Appendix A. Hamiltonian monodromy
Consider an integrable Hamiltonian system F = ( F = H, F , . . . , F n ) ona 2 n -dimensional symplectic manifold ( M, ω ). If the fibers of the integralmap F are compact and connected , then according to the classical Arnol’d-Liouville theorem [1] a tubular neighborhood of each regular fiber is a trivialtorus bundle D n × T n admitting action-angle coordinates. Hence F : F − ( R ) → R, where R ⊂ image( F ) is the set of regular values of F , is a locally trivialtorus bundle. This bundle is, however, not necessary globally trivial evenfrom the topological viewpoint. One geometric invariant that measures thisnon-triviality was introduced by Duistermaat in [12] and is called Hamil-tonian monodromy . Specifically, Hamiltonian monodromy is defined as arepresentation π ( R, ξ ) → Aut H ( F − ( ξ )) (cid:39) GL( n, Z )of the fundamental group π ( R, ξ ) in the group of automorphisms of theinteger homology group H ( F − ( ξ )) (cid:39) Z n . Each element γ ∈ π ( R, ξ ) actsvia parallel transport of integer homology cycles [12].Since the pioneering work of Duistermaat, Hamiltonian monodromy andits quantum counterpart [7, 49] have been observed in many integrable sys-tems of physics and mechanics. General results are known that allow tocompute this invariant in specific examples. It has been shown in [37, 41, 54]that in the typical case of n = 2 degrees of freedom non-trivial Hamiltonianmonodromy is manifested by the presence of the so-called focus-focus pointsof the map F . In the case of a global circle action Hamiltonian monodromy(and, more generally, fractional monodromy [42]) can be computed in termsof the singularities of the circle action [17, 38]. Remark A.1.
A notion of monodromy can be defined for torus bundlesthat do not necessarily come from an integrable system and also in the caseof bundles with non-compact fibers (for instance, in the case of cylinderbundles). Specifically, consider a bundle F : F − ( γ ) → γ, γ = S . It can be
ULER’S TWO-CENTER PROBLEM 23 obtained from a direct product [0 , × F − ( γ ( t )) by gluing the boundariesvia a non-trivial homeomorphism f , called the monodromy of the bundle.We call this monodromy Hamiltonian if F comes from a completely inte-grable system. In this case the push-forward map f (cid:63) coincides with theautomorphism given by the parallel transport.We note that non-compact fibrations appear in the Euler problem in thecase of positive energies and in various other integrable systems. We mentionthe works [22, 35, 40] and [2, 13, 16, 53]. For systems that are both scatteringand integrable scattering monodromy and Hamiltonian monodromy coincideif the reference is given by the original Hamiltonian H . Appendix B. Bifurcation diagrams for the planar problem
In this section we give bifurcation diagrams of the planar Euler problemin the case of arbitrary strengths µ i . The computation has been performedin Section 3; more details can be found in [10, 46, 51]. Figure 5.
Bifurcation diagrams for the planar problem,generic cases | µ | (cid:54) = | µ | (cid:54) = 0. Top: attractive (left), repulsive(right). Bottom: mixed. Figure 6.
Bifurcation diagrams for the planar problem,non-generic cases | µ | = | µ | or µ µ = 0. From left to right,from top to bottom: symmetric attractive, anti-symmetric,symmetric repulsive, free flow, attractive Kepler problem, re-pulsive Kepler problem. ULER’S TWO-CENTER PROBLEM 25
The computation of Section 3 yields the following critical lines(8) (cid:96) = { g = h + µ − µ } , (cid:96) = { g = h + µ − µ } and (cid:96) = { g = h + µ } , µ = µ + µ , and the critical curves { g = µ cosh λ/ , h = − µ/ λ } , { g = ( µ − µ ) sin ν/ , h = ( µ − µ ) / ν } . Points that do not correspond to any physical motion must be removed fromthe obtained set. The resulting diagrams are given in Figs. 5 and 6. Herewe distinguish two cases: generic case when the strengths | µ | (cid:54) = | µ | (cid:54) = 0and the remaining critical cases.We note that the critical cases occur when | µ | = | µ | or when µ µ = 0.In the case µ = − µ (cid:54) = 0 the attraction of one of the centers equalizes therepulsion of the other center, making the bifurcation diagram qualitativelydifferent from the cases when − µ < µ < < µ < − µ . However, westill have the three different critical lines (cid:96) , (cid:96) and (cid:96) . In the other criticalcases collisions of the critical lines (cid:96) i occur. For instance, µ = 0 implies that (cid:96) = (cid:96) and so on. The same situation takes place in the spatial problem. Appendix C. Proof of Theorem 5.4
We shall show that the Euler problem has two natural reference Hamil-tonians when µ (cid:54) = µ and one otherwise. Theorem C.1.
Among all Kepler Hamiltonians only H r = 12 p − µ − µ r and H r = 12 p − µ − µ r are reference Hamiltonians of F = ( H, L z , G ) . In particular, the free Hamil-tonian is a reference Hamiltonian of F only in the case µ = µ .Proof. Sufficiency. Consider the Hamiltonian H r . Let G r = H r + 12 ( L − a ( p x + p y )) + a ( z + a ) µ − µ r . From Section 2.1 (see also Eq. (3)) it follows that the functions H r , L z and G r Poisson commute. This implies that any trajectory g tH r ( x ) belongs tothe common level set of F r = ( H r , L z , G r ) . For a scattering trajectory wethus get F r (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = F r (cid:18) lim t →−∞ g tH r ( x ) (cid:19) . A straightforward computation of the limit shows that also F (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = F (cid:18) lim t →−∞ g tH r ( x ) (cid:19) . The case of H r is completely analogous. Figure 7.
Kepler trajectory g tH r ( x ) in the z = z plane. Necessity.
Without loss of generality µ ≤ µ . Let H r = 12 p − µr , where r : R \ { o } → R is the distance to some point o ∈ R , be a referenceHamiltonian of F . We have to show that µ > o = o and µ = µ − µ ; µ < o = o and µ = µ − µ ; µ = 0 implies µ = µ . Case 1.
First we show that o belongs to the z axis. If this is not the case,then, due to rotational symmetry, we have a reference Hamiltonian H r with o = ( − b , , z ) for some b , z ∈ R , b (cid:54) = 0. This reference Hamiltonian H r has a trajectory t (cid:55)→ g tH r ( x ) that (in the configuration space) has the formshown in Figure 7. But for such a trajectory L z (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = 0 (cid:54) = √ h · b = L z (cid:18) lim t →−∞ g tH r ( x ) (cid:19) , where h = H r ( x ) > g tH r ( x ). We conclude that o = (0 , , b )for some b ∈ R .Next we show that bµ = a ( µ − µ ). Consider a trajectory g tH r ( x ) of H r that has the form shown in Figure 8 a . It follows from Eq. (3) that thefunction G r = H r + 12 ( L − b ( p x + p y )) + b ( z + b ) µr ULER’S TWO-CENTER PROBLEM 27
Figure 8.
Kepler trajectories in the y = 0 plane.is constant along this trajectory. Thus, for H r to be a reference Hamiltonianwe must have(9) ( G − G r ) (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = ( G − G r ) (cid:18) lim t →−∞ g tH r ( x ) (cid:19) . In the configuration space, g tH r ( x ) is asymptotic to the ray x = c, y = 0 , z ≥ t = + ∞ . The other asymptote at t = −∞ gets arbitrarily close to theray x = c, y = 0 , z ≤ c → + ∞ . It follows that Eq. (9) is equivalentto a ( µ − µ ) − bµ = bµ − a ( µ − µ ) + ε, where ε → c → + ∞ .The remaining equality b = a can be proven using a trajectory g tH r ( x )that has the form shown in Figure 8 b . Case 2.
In this case trajectories g tH r ( x ) of the repulsive Kepler Hamil-tonian H r do not project to the curves shown in Figs. 7, 8 a and 8 b . How-ever, each of these curves is a branch of a hyperbola. The ‘complementary’branches are (projections of) trajectories of H r ; see Fig. 9. If the latterbranches are used, the proof becomes similar to Case 1 . Case 3.
In this case H r generates the free motion. Let g tH r ( x ) = ( q ( t ) , p ( t )) , q ( t ) = ( c, , t ) , p ( t ) = (0 , , . Since L and ( p x , p y , p z ) are conserved, G (cid:18) lim t → + ∞ g tH r ( x ) (cid:19) = G (cid:18) lim t →−∞ g tH r ( x ) (cid:19) implies a ( µ − µ ) = a ( µ − µ ) and hence µ = µ . (cid:3) Figure 9.
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