Chaos in a generalized Euler's three-body problem
KKEK-Cosmo-0272, KEK-TH-2302
Chaos in a generalized Euler’s three-body problem
Takahisa Igata ∗ KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan (Dated: February 22, 2021)Euler’s three-body problem is the problem of solving for the motion of a particle mov-ing in a Newtonian potential generated by two point sources fixed in space. This systemis integrable in the Liouville sense. We consider the Euler problem with the inverse-squarepotential, which can be seen as a natural generalization of the three-body problem to higher-dimensional Newtonian theory. We identify a family of stable stationary orbits in the gener-alized Euler problem. These orbits guarantee the existence of stable bound orbits. Applyingthe Poincar´e map method to these orbits, we show that stable bound chaotic orbits appear.As a result, we conclude that the generalized Euler problem is nonintegrable.
I. INTRODUCTION
The Kepler problem in celestial mechanics—two bodies interact with each other by the inverse-square force law—has provided us with many insights into the motion of astrophysical bodies.This problem was generalized by Euler to a certain three-body problem—motion of a particlesubjected to the inverse-square force sourced by two fixed masses in space—which is known asEuler’s three-body problem [1].These problems have a common feature of the Liouville integrability. The Kepler problem isintegrable, i.e., there exist three constants of motion, energy, angular momentum around an axis,and the squared total angular momentum, commutable with each other by the Poisson brack-ets. As a result, the equations of motion reduce to decoupled ordinary differential equations foreach variables. In addition, there is a vector-type constant of motion in this system, the so-called Laplace-Runge-Lenz vector (see, e.g., Ref. [2]). Since all six of those constants are Poissoncommutable, the Kepler problem is superintegrable and is solved in an algebraic way. On theother hand, the Euler problem admits two constants of motion, energy and angular momentum.Additionally, there is a nontrivial constant of a second-order polynomial in momentum, the so-called Whittaker constant [3]. These three constants are Poisson commutable with each other, and ∗ [email protected] a r X i v : . [ n li n . C D ] F e b therefore, the system is integrable. The nontrivial constants in both cases are relevant to hiddensymmetry developed in general relativity—the Killing tensors (see, e.g., Ref. [4]). Though theconstants are composed by reducible Killing tensors (i.e., written as tensor products of the Killingvectors) because of the maximal symmetry in the Euclidean space, they still play an importantrole in terms of the integrability of particle dynamics.Such nontrivial constants were discovered by exploring the separation of variables of the cor-responding Hamilton-Jacobi equation. This insight was applied to the geodesic systems in theKerr spacetime, and it was found that the separation of variables of the Hamilton-Jacobi equationoccurs and there exists the Carter constant [5], which is relevant to the irreducible and nontrivialKilling tensor [6]. This discovery leads us to study particle motion in the Kerr geometry furtherin analytical ways.It is a natural but nontrivial question to ask whether integrability is preserved if the systemis generalized somehow. In generalizing an integrable Newtonian particle system to a generalrelativistic particle system, we often encounter examples where integrability breaks down. Forexample, the dynamics of a massive particle in the four-dimensional (4D) Majumdar-Papapetroudihole spacetime [7, 8]—a generalization of the Euler problem to general relativity—is nonintegrablebecause it exhibits chaotic behavior [9–15]. In other relativistic Euler’s three-body problems,chaotic orbits also appear in general [16–18]. In a recent interest in higher-dimensional gravity,the particle dynamics around a uniform circular ring in the 4D Euclidean space E is known tobe integrable [19] while its relativistic generalization, the timelike geodesics in the five-dimensional(5D) black ring spacetime, is chaotic [20].One of the generalizations of the Euler problem to higher-dimensional space was also discussedin the context of Newtonian theory, in which the spatial dimension was parameterized while thepotential still being inversely proportional to distance [21]. In this case, even if the spatial dimensionis arbitrary, there exists the same type of the Whittaker constant; in other words, the integrabilityis preserved. However, in generalization of Newtonian gravity to higher-dimensional space, thepower-law of gravitational force should be modified accordingly. The purpose of this paper is toclarify whether the integrable structure of the original Euler problem can be preserved even in sucha generalized Euler’s three-body problem.It should be noted that the timelike geodesics in the higher-dimensional Majumdar-Papapetroudihole spacetime show chaotic behavior [24]. Thus, we may also ask whether the Newtonian limitof this relativistic system recovers integrability. As in the case of the black ring, the integrable The full three-body problem in Newtonian gravity with an inverse-square potential in E was discussed in Ref. [23]. structure may be recovered in the Newtonian limit. One of our purposes is to fill in the missingpieces and clarifies the boundary of integrability.To approach this problem, we focus on stable circular massive particle orbits in the 5DMajumdar-Papapetrou dihole spacetime, which are caused by the many-body effect of thesources [26, 27]. Although it should be noted that relativistic corrections in higher dimensionscan affect particle dynamics at infinity, this fact suggests that stable circular orbits also exist inNewtonian gravity. If such orbits exist in our system, stable bound orbits inevitably appear nearthese orbits (discussed in detail below). If the system is nonintegrable, then the chaos of the stablebound orbits can be determined by the Poincar´e map method.This paper is organized as follows. In Sec. II, we consider the existence of stationary parti-cle orbits in the Newtonian gravitational potential generated by two fixed centers in E . Afterformulating a method for finding stationary orbits and determining their stability, we show thesequences of stable/unstable stationary orbits. In Sec. III, we discuss the appearance of chaos forstable bound orbits by using the Poincar´e map method. Section IV is devoted to a summary anddiscussions. II. FORMULATION
We focus on the Newtonian potential generated by two point mass sources fixed at differentpoints in E . Let r be a position vector. Let r = ∓ a denote the positions of the sources with mass M ± , respectively, where a is a nonzero constant vector. Then the mass density distribution is σ ( r ) = M + δ ( r + a ) + M − δ ( r − a ) , (1)where δ denotes the delta function. Solving the Newtonian field equation with the source term σ ( r ), we obtain a Newtonian gravitational potentialΦ( r ) = − GM + | r + a | − GM − | r − a | , (3)where G is the gravitational constant. Hereafter, the masses are assumed to be equal, M ± = M .Introduce the cylindrical coordinates ( ρ, θ, φ, z ) in which the Euclidean metric takes the formd (cid:96) = d ρ + ρ (d θ + sin θ d φ ) + d z . (4) Such phenomenon is observed, for example, in particle dynamics on the 5D black ring spacetime [28, 29]. The following convention for the Newtonian field equation is used:∆Φ( r ) = Ω Gσ ( r ) , (2)where ∆ is the Laplacian of E , and Ω = 2 π is the surface area of the unit S . Without loss of generality, we may put the point sources at z = ± a on the z axis, where a = | a | .Then the potential Φ( r ) in these coordinates is given byΦ( r ) = − GM (cid:18) r + 1 r − (cid:19) , (5)where r ± = (cid:112) ( z ± a ) + ρ . (6)Let us consider freely falling particle motion in Φ( r ). Let m be particle mass and let p be acanonical momentum conjugate with coordinates of the particle. The Hamiltonian of this mechan-ical system is given by H = | p | m + m Φ (7)= 12 m (cid:18) p z + p ρ + Q ρ (cid:19) − αm (cid:18) r + 1 r − (cid:19) , (8)where α = GM and Q is defined by Q = p θ + p φ sin θ , (9)which is a constant of motion associated with the S rotational symmetry of Φ. We use units inwhich m = 1 in what follows. The Hamiltonian H is equivalent to particle energy and takes aconstant value E . The energy conservation H = E leads to the energy equation12 ( ˙ z + ˙ ρ ) + V = E, (10) V ( ρ, z ; Q ) = Q ρ − α (cid:18) r + 1 r − (cid:19) , (11)where the dots denote the derivatives with respect to time, and the Hamilton equations, p z = ˙ z and p ρ = ˙ ρ , have been used. We call V the effective potential in what follows.We focus on stationary orbits where the ρ and z coordinates of a particle remain constant.Note that all such orbits are circular because of the S rotational symmetry of Φ. To move on thecircular orbits, a particle must stay at a stationary point of the effective potential V , where thefollowing conditions must hold: V z = α (cid:18) z + ar + z − ar − (cid:19) = 0 , (12) V ρ = − Q ρ + αρ (cid:18) r + 1 r − (cid:19) = 0 , (13) V = E, (14)where V i = ∂ i V ( i = z, ρ ). The condition (12) leads to the two relations z = 0 , (15) z = ± z ( ρ ) := ± (cid:115) a − ρ a/ (cid:112) ρ + a . (16)Note that z is well-defined only in the range 0 < ρ ≤ √ a . Furthermore, we obtain the followingvalues Q and E by solving Eqs. (13) and (14) for Q and E : Q = Q := αρ (cid:18) r + 1 r − (cid:19) , (17) E = E := V ( ρ, z ; Q ) = αρ (cid:18) r + 1 r − (cid:19) − α (cid:18) r + 1 r − (cid:19) . (18)Since Q is always non-negative, we can find a stationary orbit at any point on the sequences γ := (cid:8) ( ρ, z ) \ (0 , ± a ) (cid:12)(cid:12) z = 0 or z = ± z (cid:9) . (19)Let us divide γ into two parts, depending on whether the stationary orbit at each point on γ is stable or unstable. If V at an extremum point on γ is locally minimized, then the circularorbit is stable. If not, that is, if it is locally maximized or has a saddle point, then the circularorbit is unstable. Let ( V ij ) be the Hessian matrix of V , where V ij ( ρ, z ; Q ) := ∂ j ∂ i V ( i, j = ρ, z ).Let h and k be the determinant and the trace of ( V ij ), respectively, i.e., h ( ρ, z ; Q ) := det( V ij ) and k ( ρ, z ; Q ) := tr( V ij ). Evaluating h and k at Q = Q , we obtain h ( ρ, z ) := h ( ρ, z ; Q ) = 64 α a ρ r r − − α (cid:18) r + 1 r − (cid:19) (cid:20) ( z + a ) r + ( z − a ) r − (cid:21) , (20) k ( ρ, z ) := k ( ρ, z ; Q ) = α (cid:18) r + 1 r − (cid:19) , (21)where V ij, ( ρ, z ) := V ij ( ρ, z ; Q ) are given by V ρρ, = 4 α (cid:18) r + 1 r − (cid:19) − αρ (cid:18) r + 1 r − (cid:19) , (22) V zz, = α (cid:18) r + 1 r − (cid:19) − α (cid:20) ( z + a ) r + ( z − a ) r − (cid:21) , (23) V ρz, = V zρ, = − αρ (cid:18) z + ar + z − ar − (cid:19) . (24)Note that k > γ are stable if h >
0, unstable if h <
0, and marginally stable if h = 0.We consider the stability of the circular orbits on z = z . The function h on this branchsatisfies h ( ρ, ± z ) = α a + (2 a − ρ ) (cid:112) ρ + a a ρ ( ρ + a ) / ≥ , (25) - ρ / a z / a FIG. 1. Sequences of stable/unstable circular orbits. Blue solid curves show the sequences of stable circularorbits, and blue dashed segment shows the sequence of unstable circular orbits. Orange dot denotes themarginally stable circular orbit. White circles denotes the locations of the point masses. where the equality holds for ρ = √ a . This result indicates that all the circular orbits are stableon this branch. Next we consider the stability of the circular orbits on z = 0, where h reduces to h ( ρ,
0) = 16 α a ρ − a ( ρ + a ) . (26)This implies that the circular orbits are stable (i.e., h >
0) for ρ > √ a , unstable (i.e., h < ρ < √ a , and marginally stable (i.e., h = 0) for ρ = √ a . Note that this stability behavior iscaused by switching of the stability in the z direction at ρ = √ a , V zz, ( ρ,
0) = 2 α ρ − a ( ρ + a ) . (27)The sequence of the circular orbits and their stability are shown in Fig. 1. Blue solid curves showthe sequence of stable circular orbits, and blue dashed segment shows the sequence of unstablecircular orbits.It is worth noting that while there is no stable circular orbit for a single point mass source in E , there are stable circular orbits for any nonzero value of a (up to the asymptotic region). Tosee the effect of a on the existence of stable circular orbits, we consider V on z = 0, V ( ρ,
0) = − αρ + a + Q ρ . (28)If Q ≥ α , the potential V ( ρ,
0) decreases monotonically as ρ increases, which implies that no localminimum exists. If Q < α , however, it always has a local minimum at ρ = a [ Q/ ( √ α − Q )] / ,which is caused by V → ∞ as ρ → V (cid:37) ρ → ∞ . In the limit Q → √ α , the localminimum point goes to infinity. As a result, the stable circular orbits can be interpreted as amanifestation of the many-body effect. III. STABLE BOUND ORBITS AND CHAOS
We use stable stationary orbits to find stable bound orbits—a particle moves in a spatiallybounded region without reaching infinity or the sources even if small perturbations are applied.As discussed in the previous section, a particle on a stable stationary orbit must stay at a localminimum point of the effective potential V . If the energy level E at the local minimum pointincreases slightly (i.e., some positive energy ∆ E > | ∆ E | is small enough, the effective potentialcontour at E + ∆ E will be closed. Thus, the particle with energy E + ∆ E oscillates in the vicinityof the local minimum and is bounded inside the contour, i.e., the particle moves on a stable boundorbit. Therefore, the stationary stable orbits inevitably induce the existence of stable bound orbitsin its vicinity.Figure 2 shows some typical effective potential contours in the upper panels. We use unitsin which α = 1 and a = 1 in what follows. Black solid curves denote the contours of V with Q = Q ( ρ , ρ = 2 .
0, (b) ρ = 1 .
9, and (c) ρ = 1 .
8. The position ( ρ, z ) = ( ρ , V , at which the values of E and Q are evaluated as (a)( E , Q ) = ( − . , . E , Q ) (cid:39) ( − . , . E , Q ) (cid:39) ( − . , . V = − . E = − .
025 and Q = Q ( ρ , z = 0, where ρ and p ρ are - - - - - - - - - - - - - (a) ρ = 2 (b) ρ = 1.9 (c) ρ = 1.8 FIG. 2. Typical stable bound orbits and the Poincar´e maps. Units in which α = 1 and a = 1 are used.Black solid curves in the upper panels show contours of V with Q = Q ( ρ ,
0) in the ( ρ, z ) plane, where (a) ρ = 2, (b) ρ = 1 .
9, and (c) ρ = 1 .
8. The position ( ρ, z ) = ( ρ ,
0) denotes the local minimum point of V .Red solid curves correspond to the contour level with V = − . E = − .
025 and Q = Q ( ρ , z ). The lower panels of the ( ρ, p ρ ) plane show the Poincar´e mapsof stable bound orbits for 50 random initial conditions with the same energy and the angular momentumas in the upper case. The quantities ρ and p ρ are recorded when a particle passes through the cross section z = 0 with ˙ z > recorded when a particle passes through it with ˙ z >
0. In the case (a), we find closed dotted loopsin the ( ρ, p ρ ) plane for each stable bound orbit. This implies that the orbits in the phase space lieon a torus, which means that the chaotic nature of the orbits is not manifest. In the case (b), thePoincar´e sections form closed dotted curves in some cases while most are scattered to fill boundedregions in the ( ρ, p ρ ) plane. The latter behavior is a result of the chaotic nature of stable boundorbits. In the case (c), the point set that forms a closed dotted curve no longer appears, and allorbits show chaos. Therefore, we can conclude that our generalized Euler’s three-body problemshows chaos in general. IV. SUMMARY AND DISCUSSIONS
We have considered the generalized Euler’s three-body problem, the dynamics of a freely fallingparticle in the Newtonian gravitational potential generated by point masses fixed, respectively, attwo different points in E . We presented the conditions under which particles remain in stationaryorbits in terms of the effective potential and proposed a systematic procedure for solving them.We used it to identify the sequences of stationary circular orbits in the generalized Euler problem.Furthermore, the linear stability of each circular orbit was clarified. As a result, we have founda family of stable circular orbits extending from two point masses to infinity, whose existence isindependent of the separation between the sources. It is worth noting that stable stationary orbitsdo not exist in the single point source case, i.e., the Kepler problem in E [30]. Therefore, we canconclude that the existence of stable circular orbits in this system is caused by the many-bodyeffect of the sources.We compare our results with the sequences of stable circular orbits in the 5D Majumdar–Papapetrou dihole spacetime with equal mass M ∗ and separation a ∗ [26]. In the vicinity of eachpair of sources, we can see the difference between Newtonian gravity, where there exist sequences ofstable circular orbits up to an arbitrary neighborhood of the point masses, and general relativity,where a pair of the innermost stable circular orbits appear due to the relativistic effects. Onthe other hand, in the asymptotic region, stable circular orbits exist when the dihole separationis large ( a ∗ / √ M ∗ ≥ √ a ∗ / √ M ∗ < √ ACKNOWLEDGMENTS
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