The continuous transition of Hamiltonian vector fields through manifolds of constant curvature
aa r X i v : . [ m a t h . D S ] O c t THE CONTINUOUS TRANSITION OF HAMILTONIANVECTOR FIELDS THROUGH MANIFOLDS OF CONSTANTCURVATUREFlorin Diacu, Slim Ibrahim, and Je˛ drzej Śniatycki
Pacific Institute for the Mathematical SciencesandDepartment of Mathematics and StatisticsUniversity of VictoriaP.O. Box 1700 STN CSCVictoria, BC, Canada, V8W [email protected], [email protected], [email protected]
September 6, 2018
Abstract.
We ask whether Hamiltonian vector fields defined on spaces ofconstant Gaussian curvature κ (spheres, for κ > , and hyperbolic spheres, for κ < ), pass continuously through the value κ = 0 if the potential functions U κ , κ ∈ R , that define them satisfy the property lim κ → U κ = U , where U corresponds to the Euclidean case. We prove that the answer to this question ispositive, both in the 2- and 3-dimensional cases, which are of physical interest,and then apply our conclusions to the gravitational N -body problem. Introduction
The attempts to extend classical equations of mathematical physics from Eu-clidean space to more general manifolds is not new. This challenging trend startedin the 1830s with the work of János Bolyai and Nikolai Lobachevsky, who triedto generalize the 2-body problem of celestial mechanics to hyperbolic space, [2],[14]. This particular topic is much researched today in the context of the N -bodyproblem in spaces of constant curvature, [4], [5], [6], [7], [10], [11], [12]. But ex-tensions to various manifolds have also been pursued for PDEs, even beyond theboundaries of classical mechanics, such as for Schrödinger’s equation [1], [3], [13],and the Vlasov-Poisson system, [9], to mention just a couple.In all these problems it is important to understand whether the extension ofthe equations has any physical sense. This issue, however, is specific to each caseand cannot be treated globally. The natural generalization of the gravitational N -body problem to spheres and hyperbolic spheres, for instance, is justified throughproperties preserved by the Kepler potential, which is, for any constant value ofthe curvature, a harmonic function that keeps all bounded orbits closed, [6], [7].Other problems need different unifying justifications. But a basic aspect of common interest to all attempts of extending the dynamicsfrom Euclidean space to more general manifolds is that the classical equations arerecovered when the manifolds are flattened. In the absence of this property, anyextension is devoid of meaning. Of course, this attribute alone is not enough towarrant a specific generalization, since many extensions can manifest this quality,so a choice must be also based on other criteria. Nevertheless, checking the pres-ence of this feature is an indispensable first step towards justifying any extensionof the classical equations to more general spaces.In this note we’ve set a modest goal in the basic direction mentioned above.We would like to see whether Hamiltonian vector fields defined on spheres andhyperbolic spheres tend to the classical Euclidean vector field as the curvatureapproaches zero, assuming that the classical potential is reached in the limit ina sense that will be made precise. As we will see, the answer to this problem ispositive, which may be surprising, given the fact that, from the geometrical pointof view, 2-spheres do not project isometrically on the Euclidean plane, and thesame is true for 3-spheres relative to the Euclidean space. The subtle point of theresults we obtain stays with how the limit is taken.To formalize this problem, let us consider a family of Hamiltonian vector fields,given by potential functions U κ , κ ∈ R , defined on spheres of Gaussian curvature κ > , S κ = { ( x, y, z ) | x + y + z = κ − } , embedded in the Euclidean space R , as well as hyperbolic spheres of curvature κ < , H κ = { ( x, y, z ) | x + y − z = κ − } , embedded in the Minkowski space R , , such that U κ tends to a potential U of theEuclidean plane as κ → while the Euclidean distances in R and the Minkowskidistances in R , are kept fixed when κ → . This way of approaching the limit isessential here, since allowing distances to vary with the curvature makes all thepoints of the sphere and the hyperbolic sphere run to infinity as κ → .We would like to know whether the equations of motion defined for κ = 0 tendto the classical equations of motion defined in flat space, i.e. for κ = 0 . We arealso interested in answering the same question in the 3-dimensional case, in otherwords to decide what happens when the manifolds S κ and H κ are replaced by3-spheres, S κ = { ( x, y, z, w ) | x + y + z + w = κ − } (embedded in R ), and hyperbolic 3-spheres, H κ = { ( x, y, z, w ) | x + y + z − w = κ − } (embedded in the Minkowski space R , ), respectively. Of course, we could work ingeneral for any finite space dimension, but the cases of main physical interest are he continuous transition of vector fields through manifolds of constant curvature 3 dimensions 2 and 3. Moreover, we would like to explicitly see what the equationsof motion look like in these two particular cases, such that we can use them inthe gravitational N -body problem. Another reason for including both the 2- and3-dimensional case in this note, instead of treating only the latter, is that theformer allows the reader to build intuition and make then an easy step to thehigher dimension.To address the above issues for N point masses, let us initially consider the moregeneral problem in which the bodies are interacting on a complete, connected, n -dimensional Riemannian manifold M ( n being a fixed positive integer), under alaw given by a potential function. So let N point particles (bodies) of masses m , . . . , m N > move on the manifold M . In some suitable coordinate system,the position and velocity of the body m r are described by the vectors x r = ( x r , . . . , x rn ) , ˙ x r = ( ˙ x r , . . . , ˙ x rn ) , r = 1 , N , respectively. We attach to every particle m r a metric tensor given by the matrix G r = ( g rij ) , and its inverse, G − r = ( g ijr ) , at the point of the manifold M wherethe particle m r , r = 1 , N , happens to occur at a given time instant.The law of motion is given by a sufficiently smooth potential function, U : M N \ ∆ → (0 , ∞ ) , U = U ( x ) , where x = ( x , . . . , x N ) is the configuration of the particle system and ∆ representsthe set of singular configurations, i.e. positions of the bodies for which the potentialis not defined. Then U generates a Lagrangian function(1) L ( x , ˙ x ) = 12 N X r =1 n X i,j =1 m r g rij ˙ x ri ˙ x rj − U ( x ) , which corresponds to a Hamiltonian vector field. In this setting, we can derive theequations of motion on the Riemannian manifold M in the result stated below,whose proof we also provide for the completeness of our presentation. Lemma 1.
Assume that the point masses m , . . . , m N > interact on the com-plete and connected n -dimensional Riemannian manifold M under the law im-posed by Lagrangian (1) . Then the system of differential equations describing themotion of these particles has the form (2) m r ¨ x rs = − n X i =1 g sir ∂U∂x ri − m r n X l,j =1 Γ s,rlj ˙ x rl ˙ x rj , s = 1 , n, r = 1 , N , where (3) Γ s,rlj = 12 n X i =1 g sir (cid:18) ∂g ril ∂x rj + ∂g rij ∂x rl − ∂g rlj ∂x ri (cid:19) , s = 1 , n, Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki are the Christoffel symbols corresponding to the particle m r , r = 1 , N .Proof. Recall first that g rij = g rji and g ijr = g jir for i, j = 1 , n . A simple computationleads to ∂L∂x ri = 12 n X l,j =1 m r ∂g rlj ∂x ri ˙ x rl ˙ x rj − ∂U∂x ri , ∂L∂ ˙ x ri = n X l =1 m r g ril ˙ x rl ,ddt (cid:18) ∂L∂ ˙ x ri (cid:19) = n X l =1 m r g ril ¨ x rl + n X l,j =1 m r ∂g rij ∂x rl ˙ x rl ˙ x rj . Then the Euler-Lagrange equations, ddt (cid:18) ∂L∂ ˙ x ri (cid:19) = ∂L∂x ri , i = 1 , n, r = 1 , N , which describe the motion of the N particles, take the form m r n X l =1 g ril ¨ x rl + m r n X l,j =1 (cid:18) ∂g rij ∂x rl − ∂g rlj ∂x ri (cid:19) ˙ x rl ˙ x rj = − ∂U∂x ri , i = 1 , n, r = 1 , N . Let us fix r in the above equations, multiply the i th equation by g sir , and add all n equations thus obtained. The result is m r n X i,l =1 g sir g ril ¨ x rl + m r n X i,l,j =1 g sir (cid:18) ∂g rij ∂x rl − ∂g rlj ∂x ri (cid:19) ˙ x rl ˙ x rj = − n X i =1 g sir ∂U∂x ri , r = 1 , N . Using the fact that P ni =1 g sir g ril = δ sl , where δ sl is the Kronecker delta, and theidentity n X j,l =1 ∂g rij ∂x rl ˙ x rl ˙ x rj = 12 n X j,l =1 (cid:18) ∂g ril ∂x rj + ∂g rij ∂x rl (cid:19) ˙ x rl ˙ x rj , which is easy to prove by expanding the double sums, the above equations becomethose given in the statement of the lemma, a remark that completes the proof. (cid:3) The 2-dimensional case
In this section we will prove the continuity of Hamiltonian vector fields throughthe value κ = 0 of the curvature, i.e. when we move from S κ to H κ through R . Forthis purpose, let us first define some trigonometric functions that unify circularand hyperbolic trigonometry, namely the κ -sine function, sn κ , as sn κ s := κ − / sin κ / s if κ > s if κ = 0 | κ | − / sinh | κ | / s if κ < , he continuous transition of vector fields through manifolds of constant curvature 5 the κ -cosine function, csn κ , as csn κ s := cos κ / s if κ >
01 if κ = 0cosh | κ | / s if κ < , as well as the functions κ -tangent, tn κ , and κ -cotangent, ctn κ , as tn κ s := sn κ s csn κ s and ctn κ s := csn κ s sn κ s , respectively. The following relationships, which will be useful in subsequent com-putations, follow from the above definitions: κ sn κ s + csn κ s = 1 ,dds sn κ s = csn κ s and dds csn κ s = − κ sn κ s. Also notice that all the above unified trigonometric functions are continuous rel-ative to κ .We can now prove the following result. Theorem 1.
Consider the point masses (bodies) m , . . . , m N on the manifold M κ (representing S κ or H κ ), κ = 0 , whose positions are given by spherical coordinates ( s r , ϕ r ) , r = 1 , N . Then the equations of motion for these bodies are (4) ¨ s r = − ∂U κ ∂s r + ˙ ϕ r sn κ s r csn κ s r ¨ ϕ r = − sn − κ s r ∂U κ ∂ϕ r − s r ˙ ϕ r ctn κ s r , r = 1 , N , where U κ : ( M κ ) N \ ∆ κ → (0 , ∞ ) , κ = 0 , represent the potentials, with the sets ∆ κ , κ ∈ R , corresponding to singular con-figurations. Moreover, if lim κ → U κ = U while the Euclidean distances in R (andthe Minkowski distances in R , ) between the North Pole N = (0 , , and thebodies are kept fixed, where U : ( R ) N \ ∆ → (0 , ∞ ) is the potential in the Euclidean case and ∆ is the corresponding set of singularconfigurations, then system (4) tends to the classical equations (5) ¨ x r = − ∂U ∂x r , ¨ y r = − ∂U ∂y r , r = 1 , N . Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki
Proof. In M κ , the equations of motion (2) for the N -body system take the form (6) ¨ x r = − g r ∂U κ ∂x r − g r ∂U κ ∂x r − Γ ,r ( ˙ x r ) − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − Γ ,r ( ˙ x r ) ¨ x r = − g r ∂U κ ∂x r − g r ∂U κ ∂x r − Γ ,r ( ˙ x r ) − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − Γ ,r ( ˙ x r ) , r = 1 , N . Consider now the ( s, ϕ ) -coordinates on M κ , i.e. take in system (6) thevariables s r := x r , ϕ r := x r , r = 1 , N , given in terms of extrinsic ( x, y, z ) -coordinates by x r = sn κ s r cos ϕ r , y r = sn κ s r sin ϕ r , z r = | κ | − / csn κ s r − | κ | − / , r = 1 , N . Notice that for κ → , we have z → and the remaining relations give the polarcoordinates in R .To make precise how we take the above limit, let us remark that while keepingthe bodies fixed as κ → , the geodesic distances s r , r = 1 , N , are not fixed.Indeed, we can write s r = | κ | − / α r , where α r is the angle from the centre ofthe sphere that subtends the arc of geodesic length s r . The Euclidean/Minkowskidistance corresponding to this arc is given by τ r := 2sn κ ( α r / , so we can concludethat s r = 2 sn − κ ( τ r / . The quantity τ r is assumed fixed as κ varies, and s r → τ r as κ → .Differentiating the expressions given the change of coordinates we obtain dx r = csn κ s r cos ϕ r ds − sn κ s r sin ϕ r dϕ r ,dy r = csn κ s r sin ϕ r ds r + sn κ s r cos ϕ r dϕ r ,dz r = − σ | κ | / sn κ s r ds r , where σ = 1 for κ > , but σ = − for κ < . Using these expression of thedifferentials, we can compute the line element dx r + dy r + σdz r and obtain thatthe metric tensor and its inverse are given by matrices of the form G r = ( g rij ) = (cid:18) κ s r (cid:19) , G − r = ( g ijr ) = (cid:18) κ s r (cid:19) , respectively, and that they cover the entire spectrum of metrics for κ ∈ R . So weobtain that g r = 1 , g r = g r = 0 , g r = sn κ s r , g r = 1 , g r = g r = 0 , g r = 1sn κ s r . Using the matrices G and G − as well as equations (3), we compute the Christoffelsymbols and obtain Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = 0 , Γ ,r = − sn κ s r csn κ s r , Γ ,r = Γ ,r = ctn κ s r . he continuous transition of vector fields through manifolds of constant curvature 7 Using the above results and reintroducing the index r to point out the dependenceof the equations of motion on the position of each body, system (6) becomes(7) ¨ s r = − ∂U κ ∂s r + ˙ ϕ r sn κ s r csn κ s r ¨ ϕ r = − sn − κ s r ∂U κ ∂ϕ r − s r ˙ ϕ r ctn κ s r , r = 1 , N . Given the definitions of the unified trigonometric functions at κ = 0 , the abovesystem takes in Euclidean space the form(8) ¨ s r = − ∂U ∂s r + s r ˙ ϕ r ¨ ϕ r = − s − r ∂U ∂ϕ r − s − r ˙ s r ˙ ϕ r , r = 1 , N . But, as noted above, the ( s, ϕ ) -coordinates of S κ and H κ (for κ = 0 ), becomepolar coordinates in R (for κ = 0 ), so if we write x r = s r cos ϕ r , y r = s r sin ϕ r , r = 1 , N , and perform the computations, system (8) takes the desired form (5). This remarkcompletes the proof. (cid:3) The 3-dimensional case
In this section we consider the motion in M κ , which stands for S κ or H κ . Ourmain goal is to prove the following result. Theorem 2.
Consider the point masses (bodies) m , . . . , m N on the manifold M κ (representing S κ or H κ ), κ = 0 , whose positions are given in hypersphericalcoordinates ( s r , ϕ r , θ r ) , r = 1 , N . Then the equations of motion for these bodiesare given by the system (9) ¨ s r = − ∂U κ ∂s r + ( ˙ ϕ r + ˙ θ r sin ϕ r ) sn κ s r csn κ s r ¨ ϕ r = − sn − κ s r ∂U κ ∂ϕ r + θ r sin ϕ r cos ϕ r − s r ˙ ϕ r ctn κ s r ¨ θ r = − sn − κ s r sin − ϕ r ∂U κ ∂θ r − s r ˙ θ r ctn κ s r − ϕ r ˙ θ r cot ϕ r , r = 1 , N , where U κ : ( M κ ) N \ ∆ κ → (0 , ∞ ) , κ = 0 , represent the potentials, with the sets ∆ κ , κ ∈ R , corresponding to singular con-figurations. Moreover, if lim κ → U κ = U while the Euclidean distances in R (and Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki the Minkowski distances in R , ) between the North Pole N = (0 , , , and thebodies are kept fixed, where U : ( R ) N \ ∆ → (0 , ∞ ) is the potential in the Euclidean case and ∆ is the corresponding set of singularconfigurations, then system (9) tends to the classical equations, (10) ¨ x r = − ∂U ∂x r , ¨ y r = − ∂U ∂y r , ¨ z r = − ∂U ∂z r , r = 1 , N . Proof. In M κ , system (2) takes the form(11) ¨ x r = − g r ∂U κ ∂x r − g r ∂U κ ∂x r − g r ∂U κ ∂x r − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − (Γ ,r + Γ ,r ) ˙ x r ˙ x r ¨ x r = − g r ∂U κ ∂x r − g r ∂U κ ∂x r − g r ∂U κ ∂x r − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − (Γ ,r + Γ ) ˙ x r ˙ x r − (Γ ,r + Γ ,r ) ˙ x r ˙ x r ¨ x r = − g r ∂U κ ∂x r − g r ∂U κ ∂x r − g r ∂U κ ∂x r − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − Γ ,r ( ˙ x r ) − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − (Γ ,r + Γ ,r ) ˙ x r ˙ x r − (Γ ,r + Γ ,r ) ˙ x r ˙ x r , r = 1 , N . Consider now the intrinsic ( s, ϕ, θ ) -coordinates on the manifolds M κ , i.e. take s r := x r , ϕ r := x r , θ r := x r , r = 1 , N , given in terms of the extrinsic ( x, y, z, w ) -coordinates by(12) x r = sn κ s r sin ϕ r sin θ r y r = sn κ s r sin ϕ r cos θ r z r = sn κ s r cos ϕ r w r = | κ | − / csn κ s r − | κ | − / , r = 1 , N . Notice that for κ → , we have w → and the remaining relations give thespherical coordinates in R . The remark we made in the proof of Theorem 1 (thatthe quantities s r , r = 1 , N , vary as we keep the Euclidean/Minkowski distancefixed, but tend to that distance as κ → ) applies here too without any change.Differentiating in (12), we obtain dx r = csn κ s r sin ϕ r sin θ r ds r + sn κ s r cos ϕ r sin θ r dϕ r + sn κ s r sin ϕ r cos θ r dθ r ,dy r = csn κ s r sin ϕ _ rcosθ r ds r + sn κ s r cos ϕ r cos θ r dϕ r − sn κ s r sin ϕ r sin θ r dθ r ,dz r = csn κ s r cos ϕ r ds r − sn κ s r sin ϕ r dϕ r dw r = − σ | κ | / sn κ s r ds r . he continuous transition of vector fields through manifolds of constant curvature 9 Using these expressions we can compute the line element dx r + dy r + dz r + σdw r to obtain the metric tensor and its inverse in matrix form, G r = ( g rij ) = κ s r
00 0 sn κ s r sin ϕ r , G − r = ( g ijr ) = κ s r
00 0 κ s r sin ϕ r . These matrices give all the metrics for κ ∈ R . We can thus write that g r = 1 , g r = g r = g r = g r = g r = g r = 0 , g r = sn κ s r , g r = sn κ s r sin ϕ r ,g r = 1 , g r = g r = g r = g r = g r = g r = 0 , g r = 1sn κ s r , g r = 1sn κ s r sin ϕ r . Then we can compute the Christoffel symbols and obtain Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = 0 , Γ ,r = − sn κ s r csn κ s r , Γ ,r = − sn κ s r csn κ s r sin ϕ r , Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = Γ ,r = 0 , Γ ,r = − sin ϕ r cos ϕ r , Γ ,r = Γ ,r = Γ ,r = Γ ,r = ctn κ s r , Γ ,r = Γ ,r = cot ϕ r . Then system (11) becomes(13) ¨ s r = − ∂U κ ∂s r + ( ˙ ϕ r + ˙ θ r sin ϕ r ) sn κ s r csn κ s r ¨ ϕ r = − sn − κ s r ∂U κ ∂ϕ r + θ r sin ϕ r cos ϕ r − s r ˙ ϕ r ctn κ s r ¨ θ r = − sn − κ s r sin − ϕ r ∂U κ ∂θ r − s r ˙ θ r ctn κ s r − ϕ r ˙ θ r cot ϕ r , r = 1 , N . Given the definitions of the unified trigonometric functions at κ = 0 , the abovesystem takes in Euclidean space the form(14) ¨ s r = − ∂U κ ∂s r + s r ( ˙ ϕ r + ˙ θ r sin ϕ r )¨ ϕ r = − s − r ∂U κ ∂ϕ r + θ r sin ϕ r cos ϕ r − s − r ˙ s r ˙ ϕ r ¨ θ r = − s − r sin − ϕ r ∂U κ ∂θ r − s − r ˙ s r ˙ θ r − ϕ r ˙ θ r cot ϕ r , r = 1 , N . Since the relations in (12) provide us with the spherical coordinates in R , for κ = 0 we can write that x r = s r sin ϕ r sin θ r , y r = s r sin ϕ r cos θ r , z r = s r cos ϕ r , r = 1 , N . Then straightforward computations make system (14) take the classical form (10).This remark completes the proof. (cid:3) Application to the gravitational N -body problem In this section we will apply Theorem 2 to the gravitational N -body problem.As we mentioned in the introduction, the potential we choose to define on spheresand hyperbolic spheres, which has a long history, provides the natural extension ofgravitation to spaces of constant curvature. The application of Theorem 1 wouldfollow in the same way. So we consider the cotangent potential function given by(15) U κ ( q ) = − X ≤ i The authors are indebted to NSERC of Canada for partialfinancial support through its Discovery Grants programme. References [1] V. Banica, The nonlinear Schrödinger equation on the hyperbolic space, Comm. PartialDifferential Equations , 10 (2007), 1643–1677.[2] W. Bolyai and J. Bolyai, Geometrische Untersuchungen , Hrsg. P. Stäckel, Teubner, Leipzig-Berlin, 1913.[3] N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinearSchrödinger equation on surfaces, Invent. Math. (2005), 187–223.[4] F. Diacu, On the singularities of the curved N -body problem, Trans. Amer. Math. Soc. , 4 (2011), 2249–2264.[5] F. Diacu, Polygonal homographic orbits of the curved 3-body problem, Trans. Amer. Math.Soc. (2012), 2783–2802.[6] F. Diacu, Relative equilibria of the curved N -body problem , Atlantis Studies in DynamicalSystems, vol. 1, Atlantis Press, Amsterdam, 2012.[7] F. Diacu, Relative equilibria of the 3-dimensional curved n -body problem, Memoirs Amer.Math. Soc. , 1071 (2013).[8] F. Diacu, The classical N -body problem in the context of curved space, arXiv:1405.0453. [9] F. Diacu, S. Ibrahim, C. Lind, and S. Shen, The Vlasov-Poisson System for Stellar Dynamicsin Spaces of Constant Curvature, arXiv:1506.07090, 34 p.[10] F. Diacu and S. Kordlou, Rotopulsators of the curved N -body problem, J. DifferentialEquations (2013) 2709–2750.[11] F. Diacu, E. Pérez-Chavela, and J. Guadalupe Reyes Victoria, An intrinsic approach in thecurved n -body problem, J. Differential Equations , (2012), 4529–4562.[12] F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer.Math. Soc. (2015), 1583–1593.[13] A.D. Ionescu and G. Staffilani, Semilinear Schrödonger flows on hyperbolic spaces: scat-tering in H , Math. Ann.345