Gravitational and harmonic oscillator potentials on surfaces of revolution
aa r X i v : . [ m a t h . D S ] M a y Gravitational and Harmonic Oscillator Potentialson Surfaces of Revolution
Manuele Santoprete
Department of MathematicsWilfrid Laurier University75 University Avenue West,Waterloo, ON, Canada, N2L [email protected]
Abstract
In this paper we consider the motion of a particle on a surface of revolutionunder the influence of a central force field. We prove that there are at mosttwo analytic central potentials for which all the bounded, non-singular orbitsare closed and that there are exactly two on some surfaces with constantGaussian curvature. The two potentials leading to closed orbits are suitablegeneralizations of the gravitational and harmonic oscillator potential. We alsoshow that there could be surfaces admitting only one potential that leads toclosed orbits. In this case the potential is a generalized harmonic oscillator.In the special case of surfaces of revolution with constant Gaussian curvaturewe prove a generalization of the well-known Bertrand Theorem.
PACS(2006): 45.05.+x, 45.50.Pk
I. Introduction
The problem of describing the motion of a particle on surfaces of constant curvature,under the influence of a central potential, is an interesting problem that dates backto the 19th century. Lobachevski12 was probably the first to propose an analogueof the gravitational force of Newton for the Hyperbolic space H . In 1860 Serret141eneralized the gravitational force to the sphere and solved the Kepler problem on S . In 1870 Schering16 wrote an analytical expression for the Newtonian potentialon H . Only three years later Lipschitz11 considered a one body motion in a centralpotential on the two-sphere S . In 1885 Killing7 found a generalization of all threeKepler’s laws to the case of a three-sphere S .The extension of these results to the hyperbolic case was carried out by Liebmannin 1902.9 He also derived generalizations of the oscillator potential for S and H .The well known Bertrand theorem (that states that there are only two analyticcentral potentials in Euclidean space for which all the bounded orbits are closed) wasgeneralized to the spaces S and H by Liebmann in 1903.10Many of these classical results have been long forgotten (see Ref. 15 for moredetails). However since then many authors have studied the classical Kepler problemand the quantum analogue (the hydrogen atom) rediscovering the old results andintroducing new elegant ones (see Ref. 5 for some interesting results and for anextensive bibliography on the subject).New interest on the topic was generated, at least in part, because of cosmologicalmodels as the mixmaster universe13 where the spatial slices are positively curvedand are topologically three-spheres S .In this paper we study the motion of a particle on surfaces of revolution, underthe influence of a central potential. This is a generalization of the analogous problemon surfaces of constant curvature.We first generalize the gravitational potential to surfaces of revolution in twodifferent ways. The first method is viewing the gravitational potential as a solutionof the Laplace-Beltrami equation. The second one is a generalization of an approachof Appell2 (see also Ref. 1 and 4). In this case we define the gravitational potentialand the harmonic oscillator potential on surfaces of revolution relating them to theplanar case.The potential of the gravitational interaction and the harmonic oscillator on theplane have a peculiar property: they are the only potentials that have the Bertrandproperty, i.e. that generate a central field where all the bounded non-singular orbitsare closed. Note, however, that there are non-potential forces all of whose boundedorbits are closed. See for example Ref. 18 pages 79, 80. It is therefore naturalto ask weather or not the gravitational potential on surfaces of revolution lead toclosed orbits. We show that, in general this is not the case. Indeed such potentialslead to bounded orbits only on certain surfaces of revolution with constant Gaussiancurvature.Another of the main results of the paper is the proof of Bertrand’s theorem forsurfaces of revolutions with constant Gaussian curvature: we show that on certain2urfaces the only potentials for which all the bounded non-singular orbits are closedare the generalization of the gravitational and the harmonic potential. This resultgeneralizes the proof of Liebmann10 (that holds in the case of the sphere S and thehyperbolic plane H ) and Kozlov and Harin8 (that holds in the case of the sphere).Note that, while in the case of the Euclidean plane the two-sphere and the hyperbolicplane all the bounded orbits close after one “loop”, this is not true in general forsurfaces of revolution with constant Gaussian curvature. Indeed in the latter case anon-circular orbit will close after n loops (where n is an integer that depends on thesurface).Finally we prove that, for a general surface of revolution, there are at most twocentral potentials that lead to bounded closed orbits and there are exactly two onsome surfaces of constant Gaussian curvature, in which case the potentials are thegeneralization of the gravitational and the harmonic ones. It is worth noticing that oncertain surfaces (e.g. the torus) there are no potentials leading to closed orbits. Wealso show that there could be surfaces of revolution where there is only one potentialleading to closed orbits and such potential is the generalized harmonic oscillator. Wewere unable to find any explicit example of this last kind of surfaces.The proofs use a suitable generalization of a proof of the classical Bertrand’stheorem due to Tikochinsky.17 However, we were unable to obtain a proof basedon Arnol’d’s treatment of Bertrand’s theorem (see Ref. 3, Section 2.8D). The basicidea is the following: first we treat circular orbits of radius u . These are shown toexist for potentials defined on the surfaces under consideration. Next we derive acondition for closed orbits. Then, we consider small deviations from u and, using thecondition above, we expand the effective potential to the first non vanishing order.This leads to a first condition that is expressed in the form of a differential equation.Finally, we use the next two orders in the expansion of the effective potential andfind a further condition for closed orbits. The two conditions are then analyzed andused to obtain the main results in the paper.The paper is organized as follows. In the next section we write the equationsof motion of a central potential on a surface of revolution. In Sec. III, we definethe gravitational potential and the harmonic oscillator potential on a surface ofrevolution. In Sec. IV we find an expression for the Gaussian curvature of a surfaceof revolution and we prove several facts important in the case of constant curvature.In Sec. V, we write the equations of the trajectory on a surface of revolution andwe show that the gravitational potential and the harmonic potential lead to closedorbits for certain surfaces of constant Gaussian curvature. In the last section westate and prove the main results of the paper.3 I. Equations of motion
Let I be an interval of real numbers then we say that γ : I → R is a regular planecurve if γ is C and γ ′ ( x ) = 0 for any x ∈ I . Definition 1.
Let γ : I → R be a simple (no self intersections) regular planecurve γ ( u ) = (( f ( u ) , , g ( u )) on the xz -plane where f and g are smooth curves onthe interval I , with f ( u ) > in the interior of I . Let S be a surface isometricallyembedded in R that admits a parametrization x : I × R → S of the form x ( u, φ ) = ( f ( u ) cos φ, f ( u ) sin φ, g ( u )) (1) then:1. if I = [ c, d ] and f ( c ) = f ( d ) = 0 , S is a spherical surface of revolution2. if I = ( c, d ) , with −∞ ≤ c < d ≤ ∞ , S is a hyperboloidal surface of revolution3. if I = [ c, d ] and γ ( c ) = γ ( d ) with f ( c ) = f ( d ) > , γ is a closed loop and S is atoroidal surface of revolution.4. if I = [ c, d ) , with ∞ < c < d ≤ ∞ and f ( c ) = 0 , then S is a paraboloidal surfaceof revolutionIn all cases S is a surface of revolution obtained by rotating γ about the z − axis. Thecurve γ will be called the profile curve. Note that a spherical surface of revolution is isomorphic to S and that by def-inition the sets x ( c, φ ) and x ( d, φ ) reduce to single points, i.e. the north and the south poles of S. Similarly hyperboloidal, toroidal and paraboloidal surfaces of rev-olution are homeomorphic to a hyperboloid of one sheet, a torus ( S × S ) and anelliptic paraboloid, respectively. Metric singularities can occur only on spherical andparaboloidal surfaces of revolution. If S is a spherical surface of revolution metricsingularities can only occur at the north and south poles, S is smooth everywhereelse. If S is a paraboloidal surface of revolution metric singularities can occur onlyat u = c . Hyperboloidal and toroidal surfaces of revolutions do not have metricsingularities and are smooth.Throughout this paper all surfaces of revolution will be assumed to be as inDefinition 1 (i.e. they will be either spherical, hyperboloidal, toroidal or paraboloidal)and the profile curve γ is assumed to be unit speed, i.e. ( dfdu ) + ( dgdu ) = 1.For a surface of revolution S , a simple computation gives the coefficients of thefirst fundamental form, or metric tensor (subscripts denote partial derivatives): E = x u · x u = (cid:18) dfdu (cid:19) + (cid:18) dgdu (cid:19) = 1 , F = x u · x φ = 0 G = x φ · x φ = f ( u ) ,
4o that the metric (away from any singular point) is ds = E du + 2 F du dφ + G dφ = du + f ( u ) dφ . (2)Note that the parametrization is orthogonal ( F = 0) and that E φ = G φ = 0.Surfaces given by parametrizations with these properties are said to be u -Clairaut .The Lagrangian function of a particle of mass m moving on the surface takes theform L = m u + f ( u ) ˙ φ ) − V ( u, φ )where V ( u, φ ) is the potential energy. We now consider the case where V is a functionof u alone, i.e. it is a central potential . Furthermore we assume that V is analyticexcept, at most, at the points where f ( u ) = 0, where the function is allowed to havea singularity.The Hamiltonian is H = p u m + p φ mf ( u ) + V ( u )where p φ = mf ( u ) ˙ φ . Examples:
Motion on the plane: take f ( u ) = u , g ( u ) = 0 with u ∈ (0 , ∞ ). Inthis case one recovers the usual central force problem.Motion on the sphere: take f ( u ) = sin( u ), g ( u ) = cos( u ) with u ∈ [0 , π ].Equations of motion: ˙ u = ∂H∂p u = p u m ˙ φ = ∂H∂p φ = p φ mf ( u ) ˙ p u = − ∂H∂u = p φ f ′ ( u ) mf ( u ) − dVdu ˙ p φ = − ∂H∂φ = 0Clearly H and p φ are constant of motions, they are in involution and the problem isintegrable by the Liouville-Arnold theorem.Since V : ( c, d ) → R is real analytic, standard results of differential equation the-ory guarantee, for any initial data ( u (0) , φ (0) , p u (0) , p φ (0)) the existence and unique-ness of an analytic solution defined on a maximal interval [0 , t ∗ ), where 0 < t ∗ ≤ ∞ .If if t ∗ < ∞ , we say the solution is singular . If the potential is singular at u = c and/or u = d this singularity induces singularities in the solution. If u ( t ) → c and/or u ( t ) → d as t → t ∗ we say that the solution experience a collision . It can be shownthat, in the problem under discussion, there are two types of singularities: collisions,and the singularities that arise when a solution reach the boundary of the surface ofrevolution in a finite time. 5 II. Gravitational and Harmonic potential for surfaces of rev-olution
In this section we generalize the gravitational and the harmonic oscillator potentialto general surfaces of revolution. We present two different ways to do so. The firstone starts from the observation that the gravitational potential is a solution of theLaplace equation. It is then natural to define the gravitational potential on a surfaceof revolution as a solution of the Laplace-Beltrami equation. The second is basedupon the work of Appell2 (see also Ref. 1 and 4) that used the central projection (orin cartographer’s jargon the gnomonic projection) to relate the motion on the planeto the motion on a sphere.
A. Laplace-Beltrami Equation
The Laplace-Beltrami equation generalizes the Laplace equation to arbitrary sur-faces. For a function V depending only on u , if the element of length is given byequation (2), the Laplace-Beltrami equation takes the form △ V ( u ) = 1 f ( u ) ∂∂u (cid:18) f ( u ) ∂V ( u ) ∂u (cid:19) = 0 (3)The solution of the Laplace-Beltrami equation is V ( u ) = a Θ( u ) (4)where a is a constant and Θ( u ) is an antiderivative of 1 /f ( u ) . To be more definitelet us assume a >
0. The parameter a plays the role of the gravitational constant.This generalizes the gravitational potential to surfaces of revolution. The analogueof the harmonic oscillator potential instead is given by V ( u ) = k Θ( u ) − . (5) B. Central Projection
Following Serret 14 Appell2 consider a system in R with the following equations ofmotion (in polar coordinates): ddτ (cid:18) ∂T p ∂ ( dr/dτ ) (cid:19) = R, ddτ (cid:18) ∂T p ∂ ( dψ/dτ ) (cid:19) = Ψ (6)6here T p is the kinetic energy of a point mass (of mass m = 1) in the plane T p = 12 (cid:18) drdτ (cid:19) + r (cid:18) dψdτ (cid:19) ! while R, Ψ stand for certain generalized forces.Similarly let T s be the kinetic energy of a point mass on the surface of revolution S . T s = 12 ( ˙ u + f ( u ) ˙ φ ) . The equations of motion are ddτ (cid:18) ∂T s ∂ ˙ u (cid:19) = U , ddτ (cid:18) ∂T s ∂ ˙ φ (cid:19) = Φ (7)Consider the transformation of coordinates and time given by r = X ( u ) = − Θ( u ) − , φ = ψ, dτ = Y ( u ) dt = ( f ( u )Θ( u )) − dt. (8)Then the equations (6) take the form of equations (7) where U = Y ( u ) R, Ψ = Y ( u )Φ . Now we can prove the following
Theorem 1.
There exists a trajectory isomorphism between the Lagrangian systemon R with central potential L p = 12 (cid:18) drdτ (cid:19) + r (cid:18) dψdτ (cid:19) ! + V ( r ) (9) and the Lagrangian system on the surface of revolution S , given by L s = 12 ( ˙ u + f ( u ) ˙ φ ) + V ( − Θ( u ) − ) (10) Proof.
Let Φ = Ψ = 0 and R = − ∂U∂r , U = − ∂V∂u = − ∂V∂r ∂r∂u = RY ( u )7n particular in the case of the Newtonian potential, then R = − ar = − aX ( u ) = − a Θ( u ) and thus U = Y ( u ) R = − af ( u ) integrating (and changing the sign) we find the potential V = a Θ( u )that coincides with the solution of the Laplace-Beltrami equation on the surface S .It is natural to consider V as the analogue of the gravitational potential. Similarlyin the case of the harmonic oscillator potential R = − ¯ kr = − kX ( u ) = ¯ k Θ( u ) − and thus U = Y ( u ) R = ¯ kf ( u ) Θ( u ) integrating (and changing the sign) we find V = − ¯ k Z duf ( u ) Θ( u ) = ¯ k u ) − = k Θ( u ) − where k = ¯ k/
2. It is natural to consider V as the analogue of the harmonic oscillatorpotential. IV. Gaussian Curvature of Surfaces of Revolution
Let x ( u, φ ) be a parametrization of the surface and let E = ( x u ∧ x φ ) · x uu √ EG − F , F = ( x u ∧ x φ ) · x uφ √ EG − F , G = ( x u ∧ x φ ) · x φφ √ EG − F be the coefficients of the second fundamental form in this parametrization. Then theGaussian curvature is given by the expression K = E G − F EG − F . (11)8or a surface of revolution the parametrization is given by (1) and thus E = − f g ′ , F =0, and G = f ′ g ′′ − g ′′ f ′ . Consequently the Gaussian curvature is K = − g ′ ( g ′ f ′′ − g ′′ f ′ ) f . It is convenient to put the Gaussian curvature in another form. By differentiating( f ′ ) + ( g ′ ) = 1 we obtain f ′ f ′′ = − g ′ g ′′ . Thus, K = − g ′ ( g ′ f ′′ − g ′′ f ′ ) f = − ( g ′ ) f ′′ + ( f ′ ) f ′′ ) f = − f ′′ f Now we want to study surfaces of revolution with constant curvature K . Therequirement of constant curvature gives us a linear differential equation to solve f ′′ = − Kf.
The solutions to this differential equation are of the form f ( u ) = Ae i √ Ku + Be − i √ Ku if K = 0 and f ( u ) = Cu + D if K = 0.Then, substituting f ( u ) into the unit speed relation f ′ ( u ) + g ′ ( u ) = 1 andsolving for g ( u ) gives g ( u ) = ± Z uu p − f ′ ( s ) ds (12) Remark 1.
Note that the only surfaces of revolution with zero constant curvatureare the right circular cylinder, the right circular cone and the plane.
Remark 2.
The sphere is obtained when
K > A = − B = i . The hyperbolicplane is obtained when K < A = − B = .Now we can prove the following Proposition 1.
The equation − f f ′′ + ( f ′ ) = b (13) is verified if and only if the surface of revolution S has constant Gaussian curvature K and either f ( u ) = Ae i √ Ku + Be − i √ Ku with AB = b / K or f ( u ) = Cu + D with C = ± b . roof. Note that (cid:18) ( f ′ ) − b f (cid:19) ′ = − f f ′ f ( − f f ′′ + ( f ′ ) − b ) (14)If − f f ′′ + ( f ′ ) = b then from Eq. (14) it follows that (cid:18) ( f ′ ) − b f (cid:19) = − K for some constant K . Consequently, since − f f ′′ + ( f ′ ) = b , f ′′ /f = − K and thecurvature is constant.On the other hand assume that f ′′ = − Kf . Then, if K = 0, f = Ae i √ Ku + Be − i √ Ku . Plugging this into − f f ′′ + ( f ′ ) = b we find the condition AB = b K . If K = 0 then f = Cu + D . Plugging into the equation we find C = b . Proposition 2.
The function f satisfies the equation f ′ ( u ) f ( u ) = − b Θ( u ) (15) for some antiderivative Θ( u ) of /f ( u ) , if and only if it satisfies the nonlineardifferential equation − f f ′′ + ( f ′ ) = b Proof. (cid:18) f ′ f (cid:19) ′ = f ′′ f − ( f ′ ) f = − b f , which implies (15) for some Θ( u ). Remark 3.
Note that there are nontrivial surfaces with constant Gaussian curvature(i.e. beside the Euclidean plane the Hyperbolic plane and the sphere). A surfacewith Gaussian curvature is K = 1 and b = 1 / A = 1 / B = 1 / f ( u ) = 1 / e iu + 1 / e − iu ) is depicted in Fig. 1(a). In this case f ( u ) satisfies − f f ′′ + ( f ′ ) = b with b = 1 /
2. A surface with Gaussian curvature K = − b = 1 / A = 1 / B = 1 / f ( u ) = − / e u + 1 / e − u ) is given in Fig.1(b). As in the previous example f ( u ) satisfies − f f ′′ + ( f ′ ) = b with b = 1 / a) (b) Figure 1: (a) A constant K = 1 surface with A = 1 / B = 1 / K = − A = − / B = 1 / V. Equation of the Trajectory
We now write the equation of the trajectory. Let p φ = 0. Then the coordinate φ varies monotonically and can be used as a new time. Let us put ρ = 1 /r = − Θ( u )where Θ( u ) is the antiderivative of 1 /f ( u ) selected in Proposition 2. This changeof variable has a long and distinguished history that goes back to A.C. Clairaut’s Th´eorie de la Lune (1765) and it seems strictly related to the various proofs ofBertrand’s theorem. For instance the proofs in Refs. 3,6 and the original proof ofBertrand use the change of variable above.Since p φ = mf ( u ) ˙ φ it is clear that˙ ρ = − ˙ uf ( u ) , dρdφ = − m ˙ up φ , d ρdφ = − m ¨ uf ( u ) p φ . Consequently the equation of motion¨ u = p φ dfdu m f ( u ) − m dVdu = 0can be rewritten as d ρdφ + dfdu f ( u ) − + mp φ dV (1 /ρ ) dρ = 0 (16)11f f ( u ) satisfies Eq. (13) then by Proposition 2 we have dfdu f ( u ) − = b ρ . Conse-quently we obtain d ρdφ + b ρ + mp φ dV (1 /ρ ) dρ = 0 . (17)This is the equation of the trajectory. In the case of the the Euclidean plane this issubstantially given in Newton’s Principia , Book I. §§ Th´eorie de la Lune (1765). See also Ref. 18 for a more accessible reference.
A. The Gravitational Case
In this section we study the motion under the influence of the potential V = a Θ( u ) = − aρ. In this case the equation of the trajectory (17) takes the form d ρdφ + b ρ − amp φ = 0 . The solution is given by the sum of the solution of the homogeneous equation of theform ρ = ep cos[ b ( φ − φ )] plus a solution of the non-homogeneous equation ρ = p .The solution ρ = p corresponds to the “circular orbit” of radius p = b p φ am . Consequently the trajectory is given by ρ = 1 p (1 + e cos[ b ( φ − φ )]) . B. The Harmonic Oscillator Case
In this section we study the motion under the influence of the potential V = k Θ = kρ In this case the equation of the trajectory (17) takes the form d ρdφ + b ρ − kmp φ ρ = 0 .
12 first integral of the equation above is h = 12 ( dρdφ ) + kmp φ ρ + 12 b ρ . Consequently the orbital equation is dρdφ = ± vuut h − kmp φ ρ − b ρ ! and thus φ − φ = Z ρ ( φ ) ρ dρρ r (cid:16) hρ − kmp φ − b ρ (cid:17) = 12 b Z ρ ( φ ) ρ dρρ r − (cid:0) ρ − hb (cid:1) + (cid:16) h b − kmp φ b (cid:17) . The substitution w = ρ − hb yields φ − φ = 12 b Z ww dw p − w + η where η = (cid:16) h b − kmp φ b (cid:17) and w = ρ − hb . Consequently, choosing ρ = hb , we obtain φ − φ = − b arccos (cid:18) wη (cid:19) and the equation of the orbit is given by ρ = hb + η cos[2 b ( φ − φ )] Lemma 1.
All the bounded orbits given by the gravitational and harmonic oscillatorpotential on the surface of revolution S are closed if − f ′′ + ( f ′ ) = b where b is arational number.Proof. In the case of the gravitational potential ρ = p (1 + e cos[ b ( φ − φ )]) and thebounded orbits are clearly closed if b is rational. Similarly, in the case of the harmonicoscillator, ρ = hb + η cos[2 b ( φ − φ )] and all the bounded orbits are closed providedthat b is a rational number. 13 a) (b) Figure 2: (a) A periodic orbit on a constant K = 1 surface with A = 1 / B = 1 /
8. (b) A periodic orbit on a constant K = − A = − / B = 1 / K = 1 (with A = 1 / B = 1 /
8) is depicted in Fig. 2(a). Aperiodic orbit of the generalized gravitational potential on a surface with constantcurvature K = − A = 1 / B = 1 /
8) is depicted in Fig. 2(b). Fig 2(a)and 2(b) depict examples of surfaces where all the orbits of the generalized potentialare closed. In those examples b = 1 / VI. Main Results
In this section we obtain the main results of the paper. In order to do that we needsome definitions and several lemmas. Let W ( u ) = l mf ( u ) + V ( u )with l = p φ denote the effective potential. Given the energy E and the angularmomentum l the orbit can be calculated from φ ( u ) = φ ( u ) + Z uu lmf ( u ) du q m [ E − W ( u )] .
14o prove the main theorems we first treat circular orbits of fixed radius u . Thenwe perform a first order (Lemma 4) and a third order (Lemma 5) study of the orbitswhich remain close to the circular one u . But the existence of such orbits mustfirst be guaranteed. In order to do that we first have to show that stable periodicorbits exist for all the surfaces of revolutions and potentials we consider in Bertrand’stheorem. Lemma 2.
Consider a central potential on a surface of revolution S that has atleast one bounded non-circular orbit. A necessary condition to have all the boundednon-singular orbits closed is to have a minimum of the effective potential W ( u ) (i.e.a stable circular orbit).Proof. We consider three possible cases:a) S is a spherical surface of revolutionb) S is a hyperboloidal surface of revolutionc) S is a toroidal surface of revolutiond) S is a paraboloidal surface of revolutionCase a) We distinguish several cases. If V is continuous in [ c, d ] it is bounded and W has a local minimum in ( c, d ). This is because l mf ( u ) → ∞ as u → c , u → d and V bounded in [ c, d ] imply that W ( u ) → ∞ as u → c and u → d . Now consider thecase V is not continuous in u = c (but continuous at u = d ). If W ′ ( u ) > W ′ ( u ) = 0 at some point u . If itis a saddle then all the orbits are collisions. If it it is a local maximum then theremust be a minimum point, since W ( u ) → ∞ as u → d . The case V not continuousat c is similar. Now assume V is not continuous in u = c and u = d . If W ′ ( u ) > W ′ ( u ) < c, d ) then all the orbits are collisions. Thus W ′ ( u ) = 0 for some u in ( c, d ). If it is a saddle or a maximum then all the orbits are collisions. Hence itmust be a minimum point.Case b) If W ′ ( u ) > W ′ ( u ) < u then there are no bounded non-singular orbits except, at most, the ones asymptotic to the boundary (that are notclosed). Thus there must be an u ∗ ∈ I such that W ′ ( u ∗ ) = 0. If at u = u ∗ there is alocal maximum or a saddle point there are no bounded non-singular solutions besidesthe circular one (except at most bounded solutions asymptotic to the boundary of S ). Thus W ( u ∗ ) must be a local minimum.Case c) The surface S is compact and W ( c ) = W ( d ) Since W ( u ) is a continuousfunction on [ c, d ], differentiable on ( c, d ), it has a (local) maximum or a minimumat some u ∗ ∈ ( c, d ). Clearly W ′ ( u ∗ ) = 0. If W has a local maximum then there are15ounded orbits asymptotic to the periodic one and not all the bounded orbits areclosed. Thus there must be a local minimum.Case d) If W ′ ( u ) < u ∈ I all the solutions are either unbounded or singular.If W ′ ( u ) > V ( u ) is singular at u = c then all the solutions arecollisions. On the other hand if the potential is smooth at u = c then W ( u ) → ∞ as u → c , since l mf ( u ) → ∞ as u → c . Hence if the potential is smooth W ′ ( u ) cannotbe positive for every u ∈ I . Consequently there is a u ∗ ∈ I such that W ′ ( u ∗ ) = 0. Ifsuch point is a saddle point or a maximum all the solutions are either unbounded orsingular (except at most bounded orbits asymptotic to the boundary of S ).Consider a bounded motion between turning points u and u in the vicinity ofa local minimum u of the effective potential. Let ∆ φ ( E ) denote the advance ina complete journey from u to u and back to u and let W = W ( u ) be a localminimum value, then we have the following Lemma 3.
Consider a central potential on a surface of revolution S and assume theeffective potential W has a minimum at u and yields closed orbits then Z u ( W ) u ( W ) dsf ( s ) = 2 √ mlβ p W − W (18) where β is a constant such that β = π ∆ φ = pq = 0 .Proof. Since the orbit is symmetric about the direction of a turning point we have∆ φ ( E ) = 2 Z u u lmf ( u ) du q m [ E − W ( u )]= r m l (cid:20)Z W E f ( u ( W )) du ( W ) dW dW √ E − W + Z EW f ( u ( W )) du ( W ) dW dW √ E − W (cid:21) = Z EW Γ( W ) dW √ E − W (19)where Γ( W ) = r m l (cid:20) f ( u ( W )) du dW − f ( u ( W )) du dW (cid:21) = r m l ddW "Z u ( W ) a dsf ( s ) − Z u ( W ) a dsf ( s ) . W )is a special case of Abel’s equation (or Euler’s hypergeometric transformation) andcan be solved for Γ( W ) in terms of ∆ φ ( E ) as follows. Divide both sides by √ ¯ W − E and integrate over E between W and ¯ W Z ¯ WW ∆ φ √ ¯ W − E dE = Z ¯ WW Z EW Γ( W ) √ ¯ W − E √ E − W dW dE.
A change in the order of integration leads to Z ¯ WW ∆ φ √ ¯ W − E dE = Z ¯ WW Γ( W ) dW Z ¯ WW dE √ ¯ W − E √ E − W .
The last integral is elementary. Its value is π . Let W = W ( u ). Since u ( W ) = u ( W ) = u we have Z WW ∆ φ √ W − E dE = π Z ¯ WW Γ( W ) dW = πl r m Z u ( W ) u ( W ) dsf ( s ) . (20)The previous equation is valid for any bounded motion. We now write it for closedorbits. The condition for an orbit to be closed is that ∆ φ ( E ) = q/p where q and p areintegers. If ∆ φ ( E ) / π is a continuous function of E it must be constant otherwise itwould assume irrational values. Since ∆ φ , as a function of the energy is a constant,the integration in Eq. (20) can be performed to obtain Z u ( W ) u ( W ) dsf ( s ) = 2 √ mlβ p W − W ( u ) . Lemma 4.
If in a central field on a surface of revolution S all the orbits near acircular one are closed then the potential V ( u ) satisfies the differential equation V ′′ ( u ) V ′ ( u ) = 1 f ′ ( u ) f ( u ) ( β − f ′ ( u )) ) + f ′′ ( u ) f ′ ( u ) . (21) Proof.
We now Taylor expand the effective potential W ( u ) = V ( u ) + l mf ( u ) u . With the notation W ′′ ( u ) = W ′′ , u ( W ) = u + x and u ( W ) = u − y we have, to the first non-vanishing order, W − W = 12 x W ′′ + . . . = 12 y W ′′ + . . . . Hence x = y and equation (18) yields (to this order) (cid:18)Z u u dsf ( s ) (cid:19) = (cid:18) xf ( u ) (cid:19) = 4 ml β x W ′′ ( u ) (22)The minimum condition W ′ = W ′ ( u ) = V ′ ( u ) − l f ′ ( u ) mf ( u ) = 0yields l = mf ( u ) V ′ ( u ) f ′ ( u ) . (23)Substituting Eq. (23) in Eq. (22) and using W ′′ = W ( u ) ′′ = V ′′ ( u ) + V ′ ( u ) (cid:20) − f ′′ ( u ) f ′ ( u ) + 3 f ′ ( u f ( u ) (cid:21) we obtain (21).We can now show that the gravitational potential and the harmonic oscillatorpotential on a surface of revolution S are closed only on some very special surfaces,namely on certain surfaces of constant curvature. Proposition 3.
The gravitational potential V = a Θ( u ) gives closed orbits if andonly if − f ′′ f + ( f ′ ) = β , where β is a rational number. The harmonic oscillatorpotential V = k Θ( u ) − gives closed orbits if and only if − f f ′′ + ( f ′ ) = β / , where β is a rational number.Proof. Substituting Eq. (4) in Eq. (21) and simplifying we obtain − f ′′ f +( f ′ ) = β .The first part of the proof follows from Lemma 1. Similarly substituting (5) in Eq.(21) and simplifying we obtain − f ′′ f + ( f ′ ) = β /
4. The proof follows from Lemma1. The following lemma determines the possible values of β emma 5. If in a central field on a surface of revolution S all the orbits near acircular one are closed then we obtain the following equation for ββ − − f ′′ f + ( f ′ ) ) β − f ′′ ( f ′ ) f + 4( f ′′ ) f − f ′′′ f ′ ( f ) + 4( f ′ ) = 0 . (24) Proof.
We now Taylor expand the effective potential V ( u ) around its minimum u up to order four W − W = 12 x W ′′ + 16 x W ′′′ + 124 x W ′′′′ + . . . = 12 y W ′′ − y W ′′′ + 124 y W ′′′′ + . . . , and substituting the expansion y = x (1 + ax + bx + . . . ) , we find y = x (1 + ax + a x + . . . ) with a = W ′′′ / (3 W ′′ ).When this expansion for y is inserted into Eq. (18) and powers of x up to thefourth order are kept, we obtain, (cid:18)Z u u dsf ( s ) (cid:19) = x f ( u ) (cid:20) ax + (cid:18) a + 8 af ′ ( u ) f ( u ) + 8( f ′ ( u ) f ( u ) − f ′′ ( u ) f ( u ) (cid:19) x (cid:21) = 4 ml β x (cid:20) W ′′ ( u ) + 13 xW ′′′ ( u ) + 112 x W ′′′′ ( u ) (cid:21) . Hence comparing equal powers of x f ( u ) = (cid:18) ml β (cid:19) W ′′ ( u ) (25) af ( u ) = 13 (cid:18) ml β (cid:19) W ′′′ ( u ) (26)1 f ( u ) (cid:18) a + 8 af ′ ( u ) f ( u ) + 8( f ′ ( u ) f ( u ) − f ′′ ( u ) f ( u ) (cid:19) = 13 (cid:18) ml β (cid:19) W ′′′′ ( u ) (27)The first two equations give Eq. (18). The new information is contained in the thirdequation. Simplifying the expression for the derivatives with the aid of Eqs. (23)and (21) we obtain W ′′ ( u ) = V ′ ( u ) f ′ ( u ) f ( u ) β (28) W ′′′ ( u ) = V ′ ( u ) (cid:20) f ( u ) (cid:18) β ( f ′ ( u )) − (cid:19) + f ′′ ( u )( f ′ ( u )) f ( u ) (cid:21) β (29) W ′′′′ ( u ) = V ′ ( u ) f ′ f (cid:20) β ( f ′ ) − β − f ( f ′′ ) ( f ′ ) − f ′′ f + 2 f ′′′ f ( f ′ ) + 47( f ′ ) (cid:21) β (30)19hus the quantity a is given by a = 13 (cid:20) f ′ ( u ) f ( u ) (cid:18) β ( f ′ ( u )) − (cid:19) + f ′′ ( u ) f ′ ( u ) (cid:21) Inserting the last expression and (30) into Eq. (27) yields (24).We can now prove Bertrand’s theorem for surfaces of constant curvature.
Theorem 2 (Bertrand’s Theorem for Surfaces of Constant Curvature) . Consider ananalytic central field on a surface of revolution S with constant Gaussian curvaturethat has at least one bounded non-circular orbit. Assume the effective potential W ( u ) has a local minimum. Then all the bounded (non-singular) orbits are closed if andonly if − f f ′′ +( f ′ ) = β in which case the potential energy takes the form V = a Θ( u ) or − f f ′′ + ( f ′ ) = β / in which case V = k Θ ( u ) .Proof. By Lemma 2 the hypothesis of Lemma 4 and 5 are satisfied.Since the curvature is constant then f ′′ = − Kf and either f ( u ) = Cu + D or f ( u ) = Ae i √ Ku + Be − i √ Ku . In the first case from Eq. (24) it follows that β − C β + 4 C = 0 and thus either β = C or β = 4 C . In the second case β − β KAB + 64(
KAB ) = 0 and thus either β = 4 KAB or β = 16 KAB If β = C or β = 4 KAB then by Proposition 1 f ( u ) verifies the equation − f f ′′ + ( f ′ ) = β . Using Lemma 4, i.e. substituting − f f ′′ + ( f ′ ) = β into Eq.(21) yields V ′′ ( u ) V ′ ( u ) = − f ′ ( u ) f ( u )and solving the previous differential equation we obtain V = V = a Θ( u ), whereΘ( u ) is a primitive of 1 /f ( u ) .On the other hand if β = 4 C or β = 16 KAB then by Proposition 1 f ( u ) verifiesthe equation − f f ′′ + ( f ′ ) = β /
4. Using Lemma 4, i.e. substituting − f f ′′ + ( f ′ ) = β / V ′′ ( u ) V ′ ( u ) = − f ′′ ( u ) f ( u ) + f ′ ( u ) f ( u ) f ′ ( u ) . (31)The general solution of the previous equation is of the form V ( u ) = k Θ( u ) + constant.To verify it we substitute V into Eq. (31). We obtain6 k ( f ′ ( u ) + f ( u )Θ( u )( − f ( u ) f ′′ ( u ) + ( f ′ ( u )) ))Θ ( u ) f ( u ) f ′ ( u ) = 6 k (cid:16) f ′ ( u ) + β f ( u )Θ( u ) (cid:17) Θ ( u ) f ( u ) f ′ ( u ) = 020here the last equality follows from Proposition 2 with b = β / V and V do in fact lead toclosed orbits. This follows immediately from Lemma 1. Remark 4.
Note that in the statement of Theorem 2 we added the hypothesis thatthe central field on the surface S has to have at least one non-circular periodic orbit.This is because there are no bounded orbits near the circular one and therefore theproof breaks down. However there are cases where this situation arises. For examplethis condition arises when one considers the pseudosphere (i.e. a surface of revolutionwith f ( u ) = e u ) and the gravitational potential V = a Θ( u ).We can also show a little more in the case of a general surface of revolution Theorem 3.
Consider an analytic central field on a surface of revolution S that hasat least one bounded non-circular orbit. Then there are at most two analytic centralpotentials on S for which all the bounded non-singular orbits are closed. There areexactly two (i.e. V = a Θ( u ) and V = k Θ ( u ) ) if and only if h ( u ) = − f ′′ f + ( f ′ ) ≡ constant. There is at most one if h ( u ) is not identically constant and (24) is verified.In this case the potential is V = k Θ ( u ) .Proof. By Lemma 2 the hypothesis of Lemma 4 and 5 are satisfied.Equation (24) can also be written as β − − f ′′ f + ( f ′ ) ) β + 4( − f ′′ f + ( f ′ ) ) + 3 f f ′ ( − f ′′′ f + f ′ f ′′ ) = 0 . Substituting h ( u ) = − f ′′ ( u ) f ( u ) + ( f ′ ( u )) in the previous equation yields β − h ( u ) β + 4 h ( u ) + 3 f ( u ) f ′ ( u ) h ′ ( u ) = 0 . (32)Let z = β then Eq. (32) is a quadratic equation in z . Let z and z be thesolutions of such equations. Assume z and z are constant. Then, since z + z =5 h ( u ), h ( u ) must be constant. On the other hand if h ( u ) is constant z and z areconstant. This shows that Eq. (32) has exactly two solution if and only if h ( u ) isconstant. From Proposition 1 it follows that the surface of revolution S has constantGaussian curvature. Finally, from Theorem 2 it follows that the two potentials are V = a Θ( u ) and V = k Θ ( u ) .Note that equation (21) is a first order linear differential equation of the form y ′ ( u ) + α ( u ) y = 0 (33)where y ( u ) = V ( u ) and α ( u ) = f ′ f ( β − f ′ ) ) + f ′′ f ′ . The general solution isof the form y ( u ) = Ce A ( u ) where A ′ ( u ) = a ( u ). The expression ddu (cid:0) k Θ (cid:1) = − k Θ ′ Θ u ) is an antiderivative of 1 /f ( u ))gives the general solution of Eq. (33)provided h ( u ) is not identically equal to β . In fact let Ce − A ( u ) = − k/ ( f Θ )then A ( u ) = ln (cid:0) − C k f Θ (cid:1) . Differentiating A ( u ), using that Θ ′ ( u ) = 1 /f ( u ) andsimplifying we obtain A ′ ( u ) = 2 f ′ ( u ) f ( u ) + 3 f ( u )Θ( u ) = α ( u ) = 1 f ′ ( u ) f ( u )( β − f ′ ( u )) ) + f ′′ ( u ) f ′ ( u )and solving for Θ yieldsΘ( u ) = 3 f ′ ( u ) f ( u )( − β + ( f ′ ( u )) − f ′′ ( u ) f ( u )) . Therefore differentiating the expression above, substituting the result in the equationΘ ′ ( u ) = 1 /f ( u ) and simplifying we obtain Eq. (24). Thus, if f ( u ) satisfies Eq. (24)and h ( u ) is not identically equal to β , y ( u ) = − k Θ ′ Θ is a general solution of Eq. (33)and the corresponding potential is V ( u ) = k Θ ( u ) . Acknowledgments
The author acknowledges with gratitude useful discussions pertinent to the presentresearch with Alain Albouy, Ray McLenaghan and Cristina Stoica and thanks ErnestoP´erez-Chavela for bringing to his attention the problem of the motion of a particle ona sphere. The research was supported in part by a Wilfried Laurier start-up grant.
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