Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature

Publicacions Matem`atiques, Vol 42 (1998), 267–299. NON SINGULAR HAMILTONIAN SYSTEMS AND GEODESIC FLOWS ON SURFACES WITH NEGATIVE CURVATURE Ernesto A. Lacomba and J. Guadalupe Reyes Abstract We extend here results for escapes in any given direction of the configuration space of a mechanical system with a non singular bounded at infinity homogeneus potential of degree −1, when the energy is positive. We use geometrical methods for analyzing the parallel and asymptotic escapes of this type of systems. By using Riemannian geometry methods we prove under suitable conditions on the potential that all the orbits escaping in a given direction are asymptotically parallel among themselves. We introduce a conformal Riemannian metric with negative curvature in the interior of the Hill’s region for a fixed positive energy level and we consider the boundary as a singular part of the infinity. The associated geodesic flow has as solution curves those of the problem for a fixed energy. We perform the compactification of the region via the limiting directions of the geodesic flow, obtaining a closed unit disk with a quasi-complete metric of negative curvature. 1. Introduction By a classical mechanical system we understand a triple (M,g,U) where Mis a Riemannian manifold with metric g, the function K: TM →Rdefined by Kx(v)=1 2g(v,v) for all v∈TxMis called the kinetic energy of the system, and U:M→Ris a smooth function defined in Mcalled the potential of the system. We note that Kis essentially the square of the norm of the Riemannian metric, when applied to tangent vectors of the manifold M. The function E(x, v)=K x (v)+U(π(v)) is called the total energy of the system, with x∈Mand v∈TxM, which means x=π(v), where π:TM →Mis the canonical projection. The variable vis known as the velocity. This work was partially supported by CONACYT grants with numbers 1772-E9210 and 400200-5-1406PE. 268 E. A. Lacomba, J. G. Reyes If the Riemannian metric in local coordinates is given by g=(g ij) (a symmetric positive definite matrix), then, when we consider the local kinetic energy K(x, v)=1 2g ijvivj, the total energy in these local coordinates (x, v) is written E(x, v)=1 2g ijvivj+U(x). From now on we adopt the summation convention on the repeated indices in a given formula. The physical curves of a classical mechanical system are the extremal curves of the variational principle corresponding to the Lagrangian L: TM →Rdefined as L(x, v)=K x (v)−U(π(v)). In coordinates L=L(x, v) the Euler-Lagrange equations for the variational principle are written (1) d dt µ∂L ∂vk¶=∂L ∂xk,dxk dt =vk. In a neighborhood of the point (x, p)∈T∗M, we can construct a diffeomorphism T∗M→TM locally defined by (xi,p i)→(x i,gijpj)= (x i ,vi). This diffeomorphism carries the total energy function E(x, v) into a function H:T∗M→Rlocally defined by H(x, p)=1 2 g ijpipj+ U(x), and called the Hamiltonian function. The function defined locally by vi=gijpjis called the Legendre Transformation. The Legendre transformation carries the physical curves into T∗M. Those curves in local coordinates of T∗Mare the solutions of the following system of first order diferential equations: (2)        ˙xi=∂H ∂pi , ˙pi=−∂H ∂xi, known as the Hamilton equations. If (x, p)=¡ x(t),p(t) ¢is a solution curve of (2) in T∗M, we have that dH dt ¡x(t),p(t) ¢=∂H ∂x ˙x+∂H ∂p ˙p= 0, and from this we have that the Hamiltonian His constant along the solutions of (2). This is the principle of conservation of energy which shows that it is sufficient to consider the system for fixed levels of energy. Since the Legendre transformation is a diffeomorphism, the total energy Eis also a constant of motion for the physical curves or solutions of (1) in TM. In this way, we split the total system into systems defined on the energy surfaces H=hlying in the phase space T∗M, with an effective reduction in one dimension. Hamiltonian systems and geodesic flows 269 We note that the Hamiltonian system could be locally considered as defined in the tangent bundle TM instead of the cotangent bundle T∗M, because of the Legendre transformation. For the fixed energy level h, in local coordinates we obtain the energy equation h=1 2gijpipj+U(x). Since gijpipj≥0 the energy relation define the set Mh={x∈M|U(x)≤h}, the so-called Hill’s region of system for a fixed energy level hand it is the set of possible permissible configurations for this level. We assume here that Mhis a connected, simply connected subset of M. The boundary curve Γ = {x∈Mh|U(x)=h}is where the kinetic energy vanishes and is called the zero velocity manifold. By Sard’s Theorem, except for a set of values hwith Lebesgue measure zero, we can suppose Mhis a submanifold of Mwith boundary Γ, which we assume to be also connected. 1.1. Geodesic flows. When we consider a system (M,g,U) such that U≡0, the corresponding Hamiltonian has a total energy function which coincides with the Lagrangian and it is locally written E=L=1 2gijvivj. We have the variational problem 0=δZQ P g ij ˙xi˙xjdt =δZQ Pk˙xk2 gdt defined in the tangent bundle TM on all the curves γ:x=x(t) joining the points P,Q∈M. The extremal curves (which satisfy the EulerLagrange equations) will be called the geodesics in the space Mrelative to the metric g, which in local coordinates is written gij. The set of all the solutions of this variational problem is called the geodesic flow in the manifold M, and are locally those curves x=x(t) solving the following system of differential equations ¨xi=−Γi jk ˙xj˙xk, where Γi jk =1 2gi` ³∂g`k ∂xj+∂gj` ∂xk−∂gjk ∂x`´are the Christoffel numbers of the connection asociated to the metric gij ([Du], [Sp]). We can think of the geodesic flow on a Riemannian manifold as the set of physical curves of the Hamiltonian system locally written as H= 1 2gijpipjnot subject to any potential. 270 E. A. Lacomba, J. G. Reyes 1.2. Systems with two degrees of freedom. We are interested here in Hamiltonians in R2of type H=1 2gijpipj+U(x, y) having two degrees of freedom and where gij =µm10 0m2¶is a mass matrix (positive definite), that is, in Hamiltonians whose energy function in the variables (x, y, ˙x, ˙y), have the form E=1 2¡m1˙x2+m2˙y2¢+U(x, y), where U(x, y) is the potential of the system. To obtain from the Hamiltonian the energy function defined in the tangent bundle of R2, we use the Legendre transformation gijpj=˙x i . Without loss of generality, we can study the Hamiltonians such that the energy function in the variables (x, y, ˙x, ˙y) has the form E=1 2(˙x 2+˙y 2 )+ ˜ U(x, y), which is obtained from the above by a linear change of coordinates. 2. Curvature of the Mechanical System We begin this section with the Maupertuis least action principle. The proof can be seen in the references [Du], [Ar]. Theorem 1 (Maupertuis). If our system (1) is autonomous, then for a fixed energy level h, its physical solutions are the extremals of the variational problem 0=δZγ 2K ¡˙γ(t) ¢dt, defined on all the curves (γ(t),˙γ(t)) ∈TMh. If in local coordinates K¡˙γ(t)¢=1 2gij ˙γi˙γjis the kinetic energy of the system, then the physical solutions are the extremals with fixed energy h of the variational problem with Lagrangian ˜ L=2¡1 2g ij ˙xi˙xj¢=k˙xk2 g. Corresponding to the metric gij we have the Lagrangian L=pgij ˙xi˙xj, whose corresponding functional defines arc length along curves. Its extremals are parameter independent curves, but the values of the functional on extremals permit to define the arc length parameter. A simple computation gives the following standard result. Hamiltonian systems and geodesic flows 271 Proposition 1 ([Du], [Sp]).The Lagrangian ˜ L=gij ˙xi˙xjhas the same extremals as L=pgij ˙xi˙xjup to a reparametrization. This implies that the geodesics are exactly those curves extremizing arc length. We observe from Theorem 1 that for a fixed energy level h, it is sufficient to calculate the geodesic flow corresponding to the metric gij for obtaining the physical motions of problem (1); from Proposition 1 it is sufficient with studying this flow as an one-dimensional foliation in Mh. We consider now on M=Rna Lagrangian of the form L=1 2δij ˙xi˙xj−U(x), then by the Maupertuis principle we have that for a fixed energy level h, the extremals of the problem S(γ)=Zγ δ ij ˙xi˙xjdt, γ ⊂Mh⊂Rn are the physical solutions of the mechanical problem (3) ¨x=−∇U(x). On the interior int(Mh) of the Hill’s region the identity 1 = δij ˙xi˙xj 2(h−U) holds. In particular if the curve γdoes not touch the boundary Γ of Hill’s region, it is always valid. Hence S(γ)=Zγ δ ij ˙xi˙xj=Zγ (δij ˙xi˙xj)2 2(h−U)=Zγ 1 2 (δij ˙xi˙xj)2 (h−U). From Proposition 1, the Lagrangians ˜ L=δij ˙xi˙xj=1 2 (δij ˙xi˙xj)2 (h−U) and L=1 √2 δij ˙xi˙xj √h−U have the same extremals in the interior of the Hill’s region. In this way, we introduce the metric ghin the interior of the Hill’s region Mhlocally defined by (gh)ij =µ√2δij √h−U¶, 272 E. A. Lacomba, J. G. Reyes whose geodesic flow in the interior is related to the physical curves of problem (3) and their corresponding arc length parameter is `, where d` dt =³2 h−U´1/4kdγkis a reparametrization of physical time, where kdγk2=˙x 2+˙y 2 . With this we obtain the Proposition 2. For a fixed energy level h, the physical curves of the system (3) are the geodesics asociated to the Riemannian metric (gh)ij = √2δij √h−Udefined in int(Mh). Proof: Let `be the arc length as we defined before. Then d` dt = ³2 h−U´1/4pδij ˙xi˙xj, which implies that ¡d` dt ¢2=√2 √h−Uδij ˙xi˙xj. If we consider the Lagrangian associated to the metric gh: L=1 2(gh)ij ˙xi˙xj=√2 2√h−Uδij ˙xi˙xj, and denote by (0) the derivative respect to `, we have that L=√2 2 δij(xi)0(xj)0 √h−Uµd` dt¶2 =2δij(xi)0(xj)0 2(h−U)(δij ˙xi˙xj) =2δ ij(xi)0(xj)0. In the last equality we used the fact that δij ˙xi˙xj 2(h−U)= 1 in the interior of the Hill’s region. A curve x=x(`) is a geodesic for the metric gh, if and only if it satisfies the Euler-Lagrange equations: d d` µ∂L ∂(xk)0¶=∂L ∂xk. A straightforward calculation shows that on the curve x=x(`): ∂L ∂xk= 0 and d d` µ∂L ∂(xk)0¶=4δ kj(xj)00 =4(x k ) 00. This is equivalent to (xk)00 = 0, which ends the proof. Hamiltonian systems and geodesic flows 273 We say that the metric ghis the blowing up of the metric δij on the zero velocity curve Γ, because it carries that boundary into part of the “infinity” of the Hill’s region, but with a different behaviour from other points at infinity. We define the curvature of the new mechanical system (int(Mh),g h,U) as the Gaussian sectional curvature of the metric gh. For the case of two variables x1=x,x2=ywe have the following result: Theorem 2 (Curvature). Let k(x, y)be the Gaussian curvature related to the metric ghat the point (x, y)of the interior of Hill’s region Mh.IfU(x, y)is a class C2potential, then, for the fixed level energy hwe have that 1. If ∆U(x, y)>−4k∇√h−Uk2then k(x, y)<0. 2. If ∆U(x, y)=−4k∇√h−Uk2then k(x, y)=0. 3. If ∆U(x, y)<−4k∇√h−Uk2then k(x, y)>0. Here ∆= ∂ ∂x2+∂ ∂y2is the Laplacian operator and ∇denotes the gradient. Proof: We observe that (gh)ij has the form f(x, y)δij , where f(x, y)= √ 2 ph−U(x, y), and therefore ghis conformal with the planar metric δ. We use the Gauss formula (see [Du]) for the curvature: k(x, y)= −1 2f(x, y)∆ log f(x, y). A straigthforward calculation gives us ∆ log f(x, y)=(h−U)∆U+k∇Uk2 2(h−U)2, and then k(x, y)=− √ h−U 2 √ 2··(h−U)∆U+k∇Uk2 2(h−U)2¸ =−(h−U)3/2 4√2(h−U)2·"∆U+° ° ° ° ∇U √h−U° ° ° ° 2# =−1 4√2√h−U·h∆U+4k∇√h−Uk2i =−1 4p2(h−U)·h∆U+4k∇√h−Uk2i, which ends the proof. 274 E. A. Lacomba, J. G. Reyes Corollary 1. With the same conditions of the above theorem we have that 1. If ∆U(x, y)>0then k(x, y)is negative. 2. If ∆U(x, y)=0and ∇U6=0then the curvature k(x, y)is negative. 3. If 0>∆U>− k∇Uk2 h−Uthen the curvature k(x, y)is negative. Proof: From Theorem 2 we have that k(x, y)=− √ h−U 4 √ 2·(h−U)∆U+k∇Uk2 (h−U)2¸ =−1 4p2(h−U)·∆U+k∇Uk2 h−U¸, and this proves the statement, because the quantity in the last parentheses is positive in each one of the three cases. We consider now potentials U:Mh→Rnot having singularities in the sense that some of the denominators vanish as in the case of collisions between particles in Celestial Mechanics. Then, for any initial condition for the system of differential equations (3), there exists a solution in the phase space whose projection in the Hill’s region is an extremal γ(t)=¡x(t),y(t) ¢defined in some interval J⊂R. Here tdenotes a new parameter, instead of the physical time. We say that a potential U:Mh→Ris bounded at infinity if there exist two positive numbers Rand Asuch that for every (x, y)∈Mh\BR(0,0) the inequality |U(x, y)|≤Aholds, where BR(0,0) is the open ball of radius Rand center (0,0). The following result shows that all escaping geodesics have infinite arc length in the metric gh. Lemma 1. Let U:Mh→Rbe a potential of class C2, bounded at infinity, and let γbe a solution of (3) in the Hill’s region that escapes to infinity. Then, if long¡γ(t)¢is the arc length in the metric ghof curve γ at time t, from the point P0∈γ, we have that lim t→∞long¡γ(t)¢=∞. Hamiltonian systems and geodesic flows 275 Proof: Let Abe a bound at infinity for the potential U, i.e., |U(x, y)|≤ Afor all (x, y) far enough from the origin. From the chain of inequalities 0 <|h−U|≤|h|+|U|≤h+A=M 2 with M>0, we have that 0 <√h−U≤M. This implies that Zγ(t) P0pdx2+dy2 √M≤Zγ(t) P0sdx2+dy2 √h−U, or equivalently 1 √Mlongδ¡γ(t)¢≤long¡γ(t)¢, where longδ¡γ(t)¢is the arc length of γfrom P0to γ(t) in the Euclidean metric. Since γ(t) escapes, then limt→∞ longδ¡γ(t)¢=∞, which implies necessarily that limt→∞ long¡γ(t)¢=∞. We now prove that the arc length in the metric ghof any geodesic going to the zero velocity curve Γ is finite. This is related to the fact that the metric is singular in Γ, but the corresponding improper integral is convergent. Lemma 2. Let γ⊂Mhbe a geodesic such that γintersects the zero velocity curve Γat the point P0, and let Qbe an arbitrary point of γin the interior of the Hill’s region. Then the arc length of γfrom Qto P0 is finite. Proof: We consider the chain of equalities long¡γ(P0,Q) ¢=ZQ P 0sdx2+dy2 √h−U=ZQ P0s˙x2+˙y 2 √ h−Udt =√2ZQ P0 √h−U 4 √h−Udt =√2ZQ P0 4 √h−U dt. The last integral converges because there are no singularities. From the last results we see that the geodesics through a point escaping to ∞are defined for all values of the parameter in R+. On the other hand, those intersecting the zero velocity curve Γ are defined only for values of the parameter bounded from above. We conclude the following: Corollary 2. Let U:Mh→Rbe a potential of class C2, bounded at infinity, without singularities or critical points. Then the system (int(Mh),g h,U)is not geodesically complete. 282 E. A. Lacomba, J. G. Reyes In fact for the initial conditions obtained in this way there exist infinitely many solutions of (800) whose general form is ω(τ)=α+a ρ−1 τ ρ−1+∞ X m,n≥2 amnτm(ρ−1)τn, where aρ−1is an arbitrary constant and the coefficients amn in the series do depend of αand aρ−1. We remark that we have chosen ρsuch that 2<ρ<3. When we carry these solutions into Hill’s region via the inverse transformation (x, y)→¡1 x1/ρ ,y/x ¢, they take the form y=αx +aρ−1x1/ρ +∞ X m,n≥2 amnxm(1/ρ−1)x1−n/ρ having a limiting slope at infinity equal to α, but they are not all asymptotical to a straight line of the form y=αx +βas x→∞. When we fix the direction α, the aρ−1is an arbitrary constant of integration not depending of the initial conditions ω(0) = α,β= 0 of the diferential equation (7) (see [F]). Then aρ−1generate a one-parameter family of solutions on the Hill’s region which escape in the direction α. These results rederive constructively the one obtained by Lacomba in [L2] on the existence of a two-dimensional submanifold formed by orbits which escape in each direction α, and all the projected solutions escaping in this direction can be written in this way: Theorem 4. Let Ube a homogeneous potential of degree −1, and let us fix the energy h>0. Then, for a given direction αof escape in the configuration space, there exists a two dimensional submanifold in the phase space formed by solutions of (4), which when projected into the Hill’s region become parallel or asymptotic to the straight line y=αx as t→∞. We note that in this case we impose the unboundedness condition on the Hill’s region and no topological condition; the direction on which there are parallel escapes are constrained to the possible escapes in the Hill’s region: the blowing up used for studying those escapes must be able to detect such constraints in particular problems. We refine the above result in the following section, by imposing suitable conditions on the potential U. In that case, escape in any direction will always be in such a way that the geodesics become asymptotically parallel among themselves. In fact, the above two dimensional submanifold projects nicely into a foliation of (int Mh). We give examples of this in Section 5. Hamiltonian systems and geodesic flows 283 4. Compactification of the Hill’s region From now on we assume that the potential U(x, y) satisfies any one of the three conditions in Corollary 1, so that the conformal metric ghhas negative curvature for a fixed energy level. We also assume that Uis homogeneous of degree −1, so that escape geodesics have limiting directions, as shown in Section 3. For any negative degree of homogeneity of the potential any escape solution has an asymptotic direction at infinity. This can be checked as in [L2], so the results below can be extended to this general case. We recall that the zero velocity curve is defined by Γ = {(x, y)∈ Mh|U(x, y)=h}and is the boundary of the Hill’s region. We recall also that classical mechanical systems are reversible. This means that if γ(t) is the projection on the Hill’s region of a solution curve in the phase space which intersects the curve Γ at time t= 0, then for any time twe have that γ(t)=γ(−t) which can be verified in (4) by a simple substitution. Hence, we can contruct a local one-dimensional foliation of a neighborhood of the zero velocity curve Γ contained in Mh, formed by the projections of the local solutions in phase space which touch it. This is because of the uniqueness of solutions with respect to initial conditions on Γ. In fact, the above one dimensional foliation of geodesics touching Γ is global. This is proved in Theorem 5. We study now the flow of problem (4) as the geodesic flow in int(Mh) associated to the metric gh, parametrized by arc length. We begin with the following result. Theorem 5. For any point P∈int(Mh)there exists one and only one geodesic curve through Pwhose projection intersects the zero velocity curve Γ. Proof: (Existence): Let Pbe an arbitrary point in int(Mh). If there is no such a curve, then all curves through Pescape, because there are neither non trivial critical points nor periodic orbits (Gauss-Bonnet Theorem for manifolds with negative curvature). By Lemma 1 all of them are defined for all the values of the parameter t∈R, which implies that the system is complete in P. Therefore, int(Mh) is complete itself because it is connected and simply connected. This is not posible because there are curves begining in the zero velocity curve and from Lemma 2 the points on that curve have finite distance from Γ. Then there exists a geodesic which we denote by γPconverging to Γ. From Gauss-Bonnet Theorem again, γPintersects any geodesic of the local foliation of a neigborhood of Γ in at most one point, and therefore γPdoes not acumulate in Γ, and is one of the leaves of the local foliation. 284 E. A. Lacomba, J. G. Reyes (Uniqueness): If there are two geodesics through Pconverging to the zero velocity curve in finite time, it follows from Gauss-Bonnet that any geodesic between them converges also to the zero velocity curve. Let us refer to such set of geodesics as a pencil with vertex P. We claim that the above mentioned pencil is bounded. If it is not the case, then we choose a point Qoutside the pencil, and consider a geodesic γQthrough Qconverging to the zero velocity curve and intersecting Γ at the point Q0. This point is contained in the unbounded pencil with vertex Pand therefore there exists a geodesic γPthrough P converging to Q0. This is not posible because of the uniqueness of the physical solution through Q0∈Γ with zero initial velocity. Since the above reasoning is symmetrical, the aforementioned pencil is bounded. Let γ2be the lower geodesic for the pencil, and γ1an arbitrary geodesic in the pencil. If the corresponding points for these geodesics in Γ are Q2 and Q1respectively, then all the intermediate geodesics from Pconverge to Γ between Q1and Q2. Now, by using continuity respect to initial conditions, we can choose a point Y∈Γ below Q2(outside the pencil) such that the unique geodesic going to Yintersects the other half part of the pencil (with vertex in P) at the point R6=P. For such a point R passes a geodesic which also passes through Pand converges to the zero velocity curve at the point R0. We consider now the pencil with vertices P,Q1,Q2, then, any geodesic reaching between Q1and Q2passes through P. Similarly, for the pencil with vertices R,R0,Y, any geodesic reaching between R0and Ypasses through R. Therefore, for any geodesic arising between R0and Q2we have that simultaneously passes through Pand passes through R. This is not possible because of the uniqueness of the solutions reaching the zero velocity curve. Therefore, there is only one geodesic passsing through P and converging to Γ in finite time. Definition. Let P∈int(Mh) be an arbitrary point. The polar neighborhood Bρ(P) of radius ρof P, is a neighborhood contained in int(Mh) such that all the points inside can be joined by a minimizing geodesic of length smaller than ρ. We obtain the following lemma on distances between arbitrary points in the interior of Hill’s region, via points in the boundary of polar neighborhoods. We define the distance from P∈int(Mh) to Γ, denoted d(P,Γ), as the distance from Pto P0, where P0is the point of intersection with Γ of the unique geodesic through Pgiven by Theorem 5. Hamiltonian systems and geodesic flows 285 Lemma 5. For any pair of points P,Q∈int(Mh)and any polar neighborhood of Pwith radious ρ<d(P,Γ), there exists one point Rin the circle Sρ(P)=∂(B ρ (P)) such that d(P,R)+d(R, Q)=d(P,Q). Proof: Since the circle Sρis compact, then there exists a point R∗∈Sρ such that d(P,R∗)+d(R ∗,Q) is minimal. Let us suppose that d(P,R∗)+d(R ∗,Q)=d(P,Q)+η for some number η>0. For ²>0 there exists a curve γsuch that d(P,Q)+²= long(γ). If we take ²=η 2, then d(P,R∗)+d(R ∗,Q)=η+ long(γ)−²=η 2+ long(γ) which implies that d(P,R∗)+d(R ∗,Q)>long(γ) contradicting the fact that at R∗we obtain the minimum over all the curves joining the points Qand P. This proves the lemma if we put R=R∗. The following result is a generalization of the Lemma 2.1.2 in Klingenberg [Kl] for non-complete connected, simply connected surfaces with negative curvature. Proposition 3. For any pair of points Pand Qin the interior of the Hill’s region there exists a minimizing geodesic joining them. Proof: Let us consider again the unique geodesic γ0through Pconverging to the zero velocity curve at the point P0, and let ˜ρ=d(P,P0)< ∞be the radius of the maximal polar neighborhood. 286 E. A. Lacomba, J. G. Reyes If Q∈γ0or Q∈B˜ρ, then there is nothing to prove, because from Lemma 2.1.2 in [Kl] the exponential map at Pis defined in B˜ρ(0) ⊂ TP(Mh), and the proposition holds on B˜ρ(P). So, we suppose Qis neither in γ0nor in Bρ. From Lemma 5 given 0<δ<˜ρ, there exists R∈Sδ(P) such that d(P,R)+d(R, Q)=d(P, Q). Let us denote by {Rδ,R 0 δ}=γ 0∩S δ(P) the points of intersection between γ0and Sδ. We claim that R6=Rδand R6=R0 δfor every 0 <δ<˜ρ. If this were not true, consider the set ∆={δ|0≤δ<˜ρ, Rδ∈γ0,and d(P, Rδ)+d(R δ,Q)=d(P,Q)}. It is clear because of the continuity of distance that ∆ is a closed non empty subset of R. Let δ∗= sup ∆. For given ²>0, let γRδ∗,Q be a curve such that ²+d(Rδ∗,Q) = long(γRδ∗,Q) and consider the point R∗=γRδ∗,Q ∩Sρ1 for some ρ1, with δ∗<ρ 1<˜ρ. From the definition of δ∗, is clear that R∗is not in γ0. Then, we consider the minimizing geodesic joining Pand R∗. Because the distances in B˜ρ(P) are attained at the minimizing geodesics, then the curve joining Pwith Rδ∗and Rδ∗with R∗can not minimize the distance between P and R∗unless it is itself a geodesic. This contradicts the uniqueness of the geodesics inside B˜ρ, because R∗does not belong to the geodesic γ0. Therefore R6=Rδfor all 0 <δ<˜ρ. Similarly R6=R0 δ. To complete the proof, we use the fact that we can extend to all the real numbers any geodesic through Pdifferent from γ0. The same analysis in the proof of Lemma 2.1.2 in [Kl] follows for this case, and we omit the details. Because of the Lemmas 1 and 2, given any geodesic γ, its maximal domain of definition has to be one of the following three types of intervals a) J=(−∞,a 0) (it reaches Γ in finite time as time increases). b) i) J=(a 0 ,+∞) (it escapes to ∞, arising from Γ). ii) J=R(it escapes if t→±∞). Hamiltonian systems and geodesic flows 287 We say that a geodesic γwith domain of definition as in a) is positively singular. We say that it is negatively singular if its domain falls in the case b) i). Otherwise if J=R, we say that it is regular. Definition. Let γ1:J1→Mh,γ2:J2→Mhbe two geodesics in the interior of Mh. We say the geodesics are positively asymptotic if a) Both converge to points in Γ in finite time, as the time increases, or b) Both escape to infinite as t→∞, and they have the same limiting direction at ∞. Similarly we define when two geodesics are negatively asymptotic. The following lemma is an immediate consequence of the last definition. Lemma 6. The positively (negatively) asymptotic relation between geodesics is an equivalence relation. In the “direction” of Γ there exists a foliation of the geodesic flow in a neighborhood of Γ formed by geodesics intersecting the zero velocity curve. Even more, from Theorem 5 such a foliation is global. In this way, when Uis a homogeneous potential of degree −1, we have that for every escape direction z= arctan(α) in the configuration space there exists an infinity of asymptotic geodesics which are determined by their limiting direction zat infinity. We prove below that those geodesics foliate int(Mh) for any given direction. We denote by z0the direction corresponding to Γ. That is, if a geodesic γ⊂Mhreaches Γ, then it has as limiting direction z0. Theorem 6. Given any direction z= arctan(α)in the Hill’s region and any point P∈int(Mh)there exists a unique geodesic γPthrough P and having positively asymptotic direction z. Analogously for the negatively asymptotic case. Proof: Let P∈int(Mh) be an arbitrary point and let zbe any given direction. If z=z0, then Theorem 5 proves the result. The other curves through Pescape to ∞.Ifz6=z 0 , then from Theorem 4 there exists a two dimensional submanifold in phase space formed by geodesics γ(t) such that all these satisfy γ(t)→∞when t→∞and have as limiting direction z. Let γz(t) be any of these geodesics such that it does not arise from Γ (by Theorem 5 there is at most one coming from Γ) and suppose that Pis not in γz. Then γzis defined for all t∈R. 288 E. A. Lacomba, J. G. Reyes If we consider the distance function d(t)=d(P,γz(t)), such a function is convex (d00(t)≥0), never vanishes, and it is of class C1(see Proposition 3.8.1 in [Kl]). Therefore, dattains its minimum value at some t=t0. From Lemma 3.8.2 in [Kl], the geodesic joining the points P and γz(t0) is orthogonal to γzat the point γz(t0). Let {n:n≥t0} be a sequence of positive integers going to infinity, and consider the infinite sequence of points {γz(n)}⊂γ z (R). For any integer nin the sequence there is a unique geodesic γn(t) joining the points Pand γz(n). If γn(0) = Pand we put ˙γz(0) = vn, then the sequence of tangent vectors {vn}⊂S 1⊂T P M hhas an accumulation point namely v.Asin Lemma 3.8.5 of Klingenberg [Kl], if we define the geodesic γP(t)asthe one satisfying the initial conditions γP(0) = Pand ˙γP(0) = v, this is the desired geodesic. Uniqueness follows as in Lemma 3.8.5 in [Kl] and we omit the details. Morever, also in the same Lemma 3.8.5 in [Kl], it is proven that the distance between the points γP(t) and γz(t) is bounded for t≥t0. That is, there exists a positive number K=d(P,γP(t0)) such that if t≥t0 then d(γP(t),γ z(t)) ≤3K−t0. From this and Theorem 4 we obtain the following result which generalizes the one on asymptotic parallel escapes in the direction of the central configurations given in [LR2] for isosceles 3-body problems, and for positive energy. Corollary 4. Any pair of geodesics γ1and γ2escaping in the same direction at infinity are asymptotically parallel. That is, for some big enough t0there exists a positive number Ksuch that if t≥t0then d(γ1(t),γ 2(t)) ≤K. The set of asymptotic geodesics at infinity in a given direction z= arctan(α) will be called the geodesic pencil in that direction. If there is no confusion we denote the pencil by the same symbol zcorresponding to its direction. For each pencil zat infinity, we denote by z+the class of geodesics reaching zat +∞, and by z−the class arising zat −∞.As foliations, z+and z−are the same object but the corresponding curves of each class have opposite directions. The singular geodesic pencil corresponding to the direction z0is the set of geodesics reaching Γ. The direction z0is singular in the sense that geodesics getting there, arrive in finite time and in the physical system they retrace themselves (because of the reversibility), while geodesics arrive at any other direction asymptotically in infinite time. Similarly to the case at ∞, we denote by z+ 0the class of geodesics arriving to Γ as Hamiltonian systems and geodesic flows 289 the time increases, and by z− 0to the class of geodesics arising from Γ as the time increases. We consider now the case where the boundary Γ is a connected curve with two asymptotic straight lines Lα,Lβhaving limiting directions zα and zβat ∞respectively. In such case, the directions zat ∞are bounded by zα<z<z β . Definition (Boundary of Mh). The geodesic pencils are called the points at infinity of Mhand the set formed by all of them is denoted by Mh(∞). This set of points at infinity is called the ideal boundary or absolute of Mh. The closure of Mh(∞) will be the set Mh(∞)=M h (∞)∪{z 0 }, and the closure of Mhwill be Mh=Mh(∞)∪int(Mh). If γ:J→Mhis a regular or negatively singular geodesic then γ(+∞) will denote the corresponding class of geodesics which contains it, and it can be identified with the limiting direction when the parameter tends to +∞. In a similar way if γis regular or positively singular, γ(−∞) will denote the corresponding class of geodesics which contains it as the parameter tends to −∞. The following result is an immediate consequence of Theorem 6. Corollary 5. If P∈int(Mh)and z∈Mh(∞), then there exists a unique geodesic γ⊂Mhdefined for any big enough positive time, such that γ(0) = Pand γ(+∞)=z + . Similarly for the case γ(0) = Pand γ(−∞)=z −. The following result is a direct consequence of Theorem 5. Corollary 6. If P∈int(Mh), then there exists a unique geodesic γ⊂Mhdefined in an upper unbounded interval, such that γ(0) = Pand γ∈z+ 0. Similarly for the case γ(0) = Pand γ∈z− 0. We shall denote by J=(a, b) any of the intervals defining the type of aforementioned geodesics of the problem. We understand that a<b could be any of the extremes of such type of intervals. For example, a=a0and b=∞for the case b) i). From Theorems 5 and 6, every direction z∈Mh(∞) defines two oriented foliations z−and z+of int(Mh), given by the corresponding classes, whose geodesics forming each one are the same ones but in opposite directions and are defined on suitable domains. Because every pencil foliates totally int(Mh), then we have obtained the following theorem. 290 E. A. Lacomba, J. G. Reyes Theorem 7. Given two points z1,z2in Mh(∞), there exists a unique geodesic γ(t)such that γ⊂z− 1and γ⊂z+ 2. Therefore, given two directions in the configuration space there exists a unique solution of (4) whose limiting directions are the given ones. This is summarized as Corollary 7. Any geodesic γ⊂Mhcan be written in a unique way γ=z− 1Tz+ 2as the intersection of two classes of geodesics. An equivalent way of stating the result in Corollary 7 is by claiming that any geodesic γlies in only two distinct geodesic pencils: a stable one z+ 2and an unstable one z− 1. Figure 1 illustrates this. γ γ(−∞) z− 1 z+ 2 γ(∞) Figure 1. Stable and unstable geodesic pencils in the compactification. We will now see which is the topology of the ideal boundary Mh(∞) when Mhis a connected, simply connected region, and Γ is the connected boundary of Mhhaving limiting directions as we mentioned before. Two classical examples of repulsive problems with these properties are given in Section 5. For J=(a, b) defining the domain of a geodesic passing through P, we assume 0 ∈J, in such a way that γ(0) = P. We define below a topology for Mh. Hamiltonian systems and geodesic flows 291 Definition. i) Let P∈int(Mh), z1,z2∈Mhbe points such that P6=z1,P6=z2. We define the angle between z1and z2from Pas angP(z1,z 2)= ang (˙γ1(0),˙γ2(0)) where γi:Ji→Mhare the geodesics going from Pto zirespectively. ii) Given P∈int(Mh), z∈Mh(∞) and ²>0, we define the open cone of radius ²with vertex at Pas the set CP(z,²)={ω∈M h| ω6=Pand angP(z,ω)<²}. From Gauss-Bonnet Theorem, and Theorems 5 and 6, and since k(x, y)≤0, any open cone is a sector with vertex P, and it is foliated by the geodesics through Phaving limiting directions contained in the interval (z−², z +²)⊂Mh(∞). In fact, if ω=γ(t0) for some geodesic through Pinside the cone and some t0∈J, then, for any increasing time twe have angP(z,ω) = angP(z,γ(t)). We endow the set Mhwith the topology generated by the open sets in int(Mh) and the set of all the open cones [Gr]. We observe that a sequence of points {ωi}⊂int(Mh) converges to z∈Mh(∞) if and only if, for every fixed point P∈int(Mh), d(P, ωi)→ d(P,z) and angP(ωi,z)→0. Here d(P,z) is defined as ∞if z6=z0and as d(P,Γ) if z=z0. An equivalent statement is that the geodesic arcs going from Pto ωiconverge to the geodesic arc from Pto z. The following result generalizes the Theorem 2.6.6 of Klingenberg [Kl] for complete manifolds with negative curvature. Lemma 7. Let P∈int(Mh)be an arbitrary point and let v0∈TPMh be such that the unique geodesic γ0through P=γ0(0) with initial velocity ˙γ0(0) = v0converges to Γat finite time as the time increases. Then the following restriction of the exponential map, which is defined by restricting the geodesic flow to fixed point P expP:TPMh−{tv0|t≥0}→M h−{γ 0 (t)|t≥0} is a diffeomorphism. Proof: From Theorem 5 and Proposition 3 the involved geodesics are defined for any positive time, and any pair of points inside int(Mh) can be joined by a minimizing geodesic. Now the proof follows as in Theorem 2.6.6 of Klingenberg [Kl], and we omit the details. In fact, since v06= 0, we can choose v0such that kv0k= 1 since γ0(t) is a minimizing geodesic from Pto Γ. 298 E. A. Lacomba, J. G. Reyes References [A] D. I. Anosov and V. I. Arnold,“Dynamical Systems I,” Springer-Verlag, Berlin, Heidelberg, 1988. [Am] M. A. Armstrong,“Basic Topology,” Mc Graw-Hill, London, New York, 1979. [Ar] V. I. Arnold,“Mathematical Methods of Classical Mechanics,” Springer-Verlag, New York, Heidelberg, Berlin, 1978. [Du] B. Dubrovin, A. Fomenko and S. P. Novikov,“Geometrie Contemporaine, Methodes et Applications,” 1st part, Ed. MIR, Moscow, 1985. [F] A. R. Forsyth,“Theory of Differential Equations,” vol. III, Dover, 1900. [Gr] M. Gromov, W. Ballman and V. Schroeder,“Manifolds of Nonpositive Curvature,” Birkh¨auser, Boston, Basel, Stuttgart, 1985. [Kl] W. Klingenberg,“Riemannian Geometry,” Walter de Gruyter ed., Berlin, New York, 1982. [L1] E. A. Lacomba, Blow up on energy levels in celestial mechanics, Publ. Mat. 28 (1984), 97–117. [L2] E. A. Lacomba, Infinity manifolds for positive energy in celestial mechanics, Contemp. Math. 58, Part III (1987), 193–201. [LBI] E. A. Lacomba, J. Bryant and L. A. Ibort, Blow up of mechanical systems with a homogeneous energy, Publ. Mat. 35 (1991), 333–345. [LPCh] E. A. Lacomba and E. A. P´ erez-Chavela, A compact model for the rhomboidal planar four-body problem, Celestial Mech. Dynam. Astronom. 54 (1992), 343–355. [LR1] E. A. Lacomba and J. G. Reyes, Parallel escapes in the restricted repulsive Coulombian isosceles 3-body problem, in “New trends for hamiltonian systems and celestial mechanics,” Advanced series in nonlinear dynamics 8, Singapore, World Scientific, 1996, pp. 260–273. [LR2] E. A. Lacomba and J. G. Reyes, Unbounded Parallel and asymptotic solutions in the isosceles 3-body problems with inverse-square forces, Report no. 130, Centre de Recerca Matem`atica, Bellaterra, Barcelona, Spain (1991). [O] C. P. Ong, Curvature and mechanics, Adv. Math. 15 (1975), 269–311. Hamiltonian systems and geodesic flows 299 [Sp] M. A. Spivak,“A Comprehensive Introduction to Differential Geometry,” vol. 1, Publish or Perish, Inc., Boston, Massachusetts, 1970. Departamento de Matem´aticas Universidad Aut´onoma Metropolitana-Iztapalapa Apartado postal 55-534 09340 M´exico D.F. M´ EXICO e-mail: [email protected] e-mail: [email protected] Primera versi´o rebuda el 8 de gener de 1997, darrera versi´o rebuda el 10 de desembre de 1997