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The Curved N-Body Problem: Risks and Rewards

FLORIN DIACU

xploring new mathematical territory can be a risky motion of a body about a fixed ) in which the force is adventure. With no landmarks in sight, it’s easy to inversely proportional to the area of a sphere of radius EE err, as all the giants, from Fermat and Newton to equal to the hyperbolic arc distance between the body and Gauss and Poincare´, well knew. Still, an expedition into the the attraction centre. Alas, they never followed this idea to unknown is highly rewarding if planned with care and obtain the equations of motion in analytic form [4, 28]. advanced with restraint, a thing the masters knew too. I One who appreciated the problem was learned this lesson not only by reading them and about Lejeune Dirichlet. He told friends he had dealt with it during them, but also from experience. his last year in Berlin (1852), but he never published anything in this direction [27]. It was Ernest Schering, a Go¨ttingen A Brief History of the Problem mathematician and reluctant editor of Gauss’s posthumous The curved N-body problem is a short name for the work, who came up with an analytic expression of the 3 equations that describe the motion of gravitating particle potential in 1870 [33]. Since the area of a sphere in H is systems in of constant Gaussian curvature. I dis- proportional to sinh2 r, where r is its radius, Schering pro- covered this universe a few years ago, and found it posed a force, F, inversely proportional to this function, fascinating. Later I also looked into its history, which which led him to a potential, U, that involves coth r, because proved to be as gripping as the universe itself. F = U. But was this the right way to extend Newton’s law? r The classical N-body problem has its roots in the work of After all, there are many ways to do that. Rudolf Lipschitz Isaac Newton, who introduced it in his masterpiece Prin- must have asked this question, for in 1873 he defined a 3 cipia in 1687. His goal was to study the motion of the moon, potential in S proportional to 1= sin r. His attempts to solve and for this he offered a model for the dynamics of celestial the Kepler problem led him to expressions that depend on objects under gravitational attraction. It took about 150 years elliptic integrals, which cannot be explicitly solved, so his until the independent founders of hyperbolic , approach was a dead end. In 1885, Wilhelm Killing returned 3 Ja´nos Bolyai and , thought of extending to the original idea and used cot r for the potential in S [22]. gravitation to spaces of constant curvature [4, 28]. Still, without knowing whether the orbits following this By that time, Gauss had already derived his law, gravitational law are conic sections, as it happens in the which is equivalent to Newton’s, but emphasizes that the Euclidean case, a question mark loomed over this extension gravitational force is inversely proportional to the area of of gravity to spaces of constant curvature. the sphere of radius equal to the distance between the The breakthrough came at the dawn of the 20th century attracting bodies. So it’s no wonder that both Bolyai and thanks to Heinrich Liebmann, a mathematician known Lobachevsky saw the connection between the geometry of today mostly for his work in . In 1902, the and the laws of physics. They defined a Kepler Liebmann published a paper in which he showed that the 3 problem in the hyperbolic space H (the gravitational orbits of the curved Kepler problem are indeed conic

24 THE MATHEMATICAL INTELLIGENCER Ó 2013 Springer Science+Business Media New York DOI 10.1007/s00283-013-9397-1 Author's personal copy sections in the hyperbolic plane and that the coth potential had been my source of inspiration, the only important is a harmonic function in 3D, but not in 2D, the same as in development was the work on the 2-body problem initiated the classical Kepler problem [25]. His next key result was by the Russian school of , led by Valery related to Bertrand’s theorem, which states that for the Kozlov [23]. These researchers discovered that, unlike in Kepler problem there are only two analytic central poten- Euclidean space, where the Kepler problem and the 2-body tials in the Euclidean space for which all bounded orbits are problem are equivalent, in curved space the equations of closed, one of the potentials being the inverse square law motion are distinct. Moreover, the curved Kepler problem is [2]. In 1903, Liebmann proved a similar theorem for the integrable, whereas the curved 2-body problem is not [34]. 2 2 cotangent potential in H and S [26]. These properties When I wanted to extend the curved 2-body problem to established the law proposed by Bolyai and Lobachevsky any number N C 2 of bodies, the only reference point I had as the natural extension of gravity to spaces of constant was the paper of the Spanish . I learned curvature, and explained why the inverse square of the arc about the work of Schering, Killing, and Liebmann only distance, which seems more natural but doesn’t have these months after discovering this new universe. But before features, is not the way to go. getting into mathematical details, I will recount how every- This great news, which meant that my research on the thing started. curved N-body problem was on the right track, also made me curious about Liebmann. I learned that he had been Beginnings briefly the rector of the University of Heidelberg, my own We feel restless without a good problem to solve, a point in doctoral Alma Mater, but had been ousted from his post life we can reach for various reasons, including loss of soon after the Nazis came to power, during their abomi- interest in what we’ve been doing. Most of us hit this wall nable purge of Jewish intellectuals. In 2009, I visited sooner or later, and so did I. Finding a new appealing Heidelberg again, and learned from my Ph.D. supervisor, subject can be challenging under such circumstances. But Willi Ja¨ger, about the Liebmann Remembrance Colloquium in this case chance favoured me. of 2008. Liebmann’s son Karl-Otto, also an academic, had I had worked in the dynamics of particle systems for more donated on this occasion an oil portrait of his father to the than 20 years. I started with the classical N-body problem university. This painting has now a place of honour in the and later introduced the Manev model and its generalization, Faculty of Mathematics. the quasihomogeneous law, which includes those of New- Not much was done after Liebmann’s breakthrough. ton, Birkhoff, Coulomb, Schwarzschild, Lennard-Jones, Van Before the recent publication of a paper by the Spanish der Waals, and a few others. The mathematical community mathematicians Carin˜ena, Ran˜ada, and Santander [3], which embraced the topic and produced some 200 papers so far. But a few years ago I felt tired of studying the same equations and ready for a new challenge...... I was teaching a course in non-, focusing on the hyperbolic plane and using synthetic FLORIN DIACU obtained his mathematics methods and complex functions in the Poincare´ and Klein- diploma from the University of Bucharest in Beltrami disks and the Poincare´ upper-half plane. I also

AUTHOR Romania and his Ph.D. degree from the used Weierstrass’s model (often wrongly attributed to Lor- 2 University of Heidelberg in Germany. He entz), the natural analogue to the sphere S of elliptic has been teaching at the University of Vic- geometry. To define it, take the upper sheet of the toria in Canada for more than two decades. Apart from mathematics research, he enjoys writing nonfiction and poetry. His latest book, Megadisasters—The Science of Predicting the Next Catastrophe, published by Princeton University Press in North America and Oxford University Press in the rest of the English-speaking world, won a ‘‘Best Academic Book Award’’ in 2011. In his leisure time he enjoys reading, playing tennis, and hiking.

Pacific Institute for the Mathematical Sciences and Department of Mathematics and Statistics University of Victoria Figure 1. Weierstrass’s model of the hyperbolic plane (given Victoria, BC by the upper sheet of the hyperboloid of two sheets) is Canada V8W 3R4 equivalent to the Poincare´ disk model, a fact that can be e-mail: [email protected] proved by stereographic projection.

Ó 2013 Springer Science+Business Media New York, Volume 35, Number 3, 2013 25 Author's personal copy

2 hyperboloid of two sheets, H x; y; z x2 y2 z2 Deriving the Equations of Motion ¼fð Þj þ À ¼ 1; z [ 0 (see Figure 1). The geodesics are the hyperbo- In 2D, the right setting for the problem is on spheres and À g las obtained by intersecting this surface with the planes that hyperbolic spheres, i.e., contain the frame’s origin. Through a point outside a 2 2 2 2 1 S x; y; z x y z jÀ and geodesic, we can draw two parallel geodesics that meet the j ¼fð Þj þ þ ¼ g 2 2 2 2 1 original one at infinity, each at a different end, so the main Hj x; y; z x y z jÀ ; z [ 0 ; of hyperbolic geometry is satisfied. ¼fð Þj þ À ¼ g Indeed, take a geodesic , i.e., a obtained by respectively, depending on the sign of the curvature j. c 2 intersecting a plane through the origin, O, of the coordinate When j? 0 we recover R (see Figure 2). These 3 system with 2. This hyperbola has two asymptotes in its are embedded in the ambient space, which is R with the H 3 plane: the straight lines a and b, intersecting at O. Take a standard inner product, for positive curvature, but R with 2;1 point, P, on the surface that is not on the chosen hyperbola. the Lorentz inner product, i.e., the Minkowski space R , The plane aP produces the geodesic hyperbola a, whereas for negative curvature. In other words, the inner product bP produces b. These two intersect at P. Then a of the vectors a = (ax, ay, az) and b = (bx, by, bz) is a b axbx ayby razbz , where r = 1 for j [ 0 and and c meet at infinity along a, whereas b and c meet at Á ¼ þ þ infinity along b. All the hyperbolas between a and b (also r =-1 for j \ 0. obtained from planes through O) are non-secant with c. The Lagrangian formulation of mechanics can be 2 3 2 obtained from a basic variational principle that minimizes a But unlike S , which is embedded in R , H is embed- 3 function called action, ded in the 3D Minkowski space, i.e., R with the Lorentz inner product, which has the (+, +, -) signature instead of t2 the standard (+, +, +). A different inner product affects A Ldt; geometric orthogonality. Indeed, a tangent vector to 2 and ¼ S Zt the radius vector touching it are orthogonal, i.e., their inner 1 2 product is zero. In H things change. The vector from the where L = T - V is the Lagrangian function, with T and origin and the corresponding tangent vector don’t look V denoting the kinetic and the potential of a certain orthogonal, but their Lorentz inner product is zero, so they mechanical system and t1 \ t2 are two instants in time. Such are to each other in the Minkowski space. a minimum leads to the differential equations that describe Moreover, this inner product is responsible for the negative the motion. 2 curvature of H , which appears positively curved to the The theory of constrained Lagrangian dynamics adds naked eye. some conditions, which in our case are those of keeping the I also happened to read at that time the paper about the bodies m1; m2; ...; mN on the [20]. For our problem, 2 2 Kepler problem in S and H by Carin˜ena, Ran˜ada, and if qi = (xi, yi, zi) are the vectors, q_ i x_i; y_i; z_i the Santander [3]. The force between the two points acts along , and the constraints are characterized¼ð byÞ the 2 the geodesic that connects them; in S , the direction is that equations fi 0; i 1; 2; ...; N , then the motion is descri- of the shortest distance, except for the antipodal case, bed by the Euler-Lagrange¼ ¼ equations with constraints, when a singularity occurs. The potential defining the 2 d oL oL ofi equations uses the cotangent of the arc distance in S and ki t 0; i 1; 2; ...; N ; 2 dt oq_ À oq À ð Þ oq ¼ ¼ the cotangent hyperbolic of the arc distance in H . As I i i i explained earlier, this potential, and not the inverse arc where the functions ki; i 1; 2; ...; N, are the Lagrange distance—as intuition suggests— is the natural extension of multipliers. ¼ the Newtonian model to curved space. To bring the positive and negative curvature cases into The Spanish mathematicians had written the equations one equation, we can follow the Spanish mathematicians of motion in intrinsic coordinates, using differential 2 and unify circular and hyperbolic trigonometry. For this, geometry, and obtained conic sections as orbits in S and 2 2 we define the j-sine as H , in agreement with what we know from R . This result gave me the idea to find the equations of the curved N- body problem, so I tried the intrinsic approach. But the computations proved complicated, so I emailed my friends Manuele Santoprete, a former student of mine, now a professor at Wilfrid Laurier University, and my old collab- orator Ernesto Pe´rez-Chavela of Mexico City, hoping that we could find a new approach. They eagerly accepted my challenge. We exchanged emails for weeks, during which a lucky combination of ideas shaped up. The insight we achieved in this way proved crucial. In the next two sec- tions I will summarize what we found during the couple of months that followed. Later, after obtaining the equations of motion, thus knowing what to expect, we also derived Figure 2. The continuous deformation of the sphere into a them in intrinsic coordinates [16, 31]. plane into a hyperbolic sphere.

26 THE MATHEMATICAL INTELLIGENCER Author's personal copy

1=2 1=2 1=2 3=4 jÀ sin j x if j [ 0 q j À r ; i 1; 2; ...; N ; and s j t; ð Þ i ¼j j i ¼ ¼j j sn x : x if 0 j 8 j and redenoting the new position vectors r by q and the ð Þ ¼ 1=2 1=2 ¼ i i <> j À sinh j x if j\0; ðÀ Þ ððÀ Þ Þ fictitious time s by t, the equations of motion can be written as: the j-cosine as :> N 1=2 mj qj r qi qj qj cos j x if j [ 0 q€ ½ À ð Á Þ Š r q_ q_ q ; q q r; ð Þ i ¼ 2 3=2 À ð i Á iÞ i i Á i ¼ csnj x : 1 if j 0 j 1;j i r r qi qj ð Þ ¼ 8 ¼ ¼X6¼ ½ À ð Á Þ Š <> cosh j 1=2x if j\0; i 1; 2; ...; N : ððÀ Þ Þ ¼ as well as the j-tangent:> and j-cotangent as 2 So for positive curvature, the problem is reduced to S , and 2 snj x csnj x for negative curvature to H . It is not difficult to see that the tnj x : ð Þ and ctnj x : ð Þ ; ð Þ ¼ csnj x ð Þ ¼ snj x equations of motion are Hamiltonian, with the energy ð Þ ð Þ integral given by respectively. The fundamental formula of unified T q; q_ U q h; trigonometry becomes ð ÞÀ ð Þ¼ j sn2 x csn2 x 1: where h is the energy constant and jð Þþ jð Þ¼ N For fun, students can try to derive other formulas. 1 3 T q; q_ mi q_ i q_ i rqi qi ; The distance between two points a; b R can be ð Þ¼2 i 1 ð Á Þð Á Þ 2 X¼ defined as N rm m q q i j i Á j U q 2 1=2 1=2 1 ja b ð Þ¼ r q q q q r q q dj a; b : rj À csnÀ Á ; j 1;j i i i j j i j ð Þ ¼ð Þ j pja apjb b ¼X6¼ ½ ð Á Þð Á ÞÀ ð Á Þ Š Á Á are the kinetic energy and force function, respectively, in -1 where csnj is the inverse trigonometricffiffiffiffiffiffiffiffiffiffiffi functionffiffiffiffiffiffiffiffiffiffiffiffi of csnj. the new coordinates. The total 2 2 On Sj and Hj this is the arc distance. For j? 0, we recover produces three more integrals: the Euclidean norm. If we further define the kinetic energy T as N miqi q_ i c; N i 1  ¼ 1 X¼ Tj q; q_ : mi q_ i q_ i jqi qi ; ð Þ ¼ 2 i 1 ð Á Þð Á Þ where c is a constant 3-vector and 9 denotes the cross X¼ product. the as V =-Uj, where The equations of motion generalize easily to any higher 3 , but the ones of physical interest so far are to S Uj q : mimj ctnj dj qi; qj 3 ð Þ ¼ 1 i\j N ð ð ÞÞ and H . For this, we introduce the coordinates 4 X (w, x, y, z) in R . With this in mind, the equations of is the force function, and the constraints with the help of motion and the energy integral have the same form as 2 2 2 1 the functions fi xi yi rzi jÀ ; i 1; 2; ...; N ; the above. But the angular momentum integrals change since ¼ þ þ À ¼ 4 equations of motion take the form the concept of cross product has no meaning in R . We need to employ the exterior wedge product, , which leads miq€ q Uj q jmi q_ q_ q ; i 1; 2; ...; N ; i ¼r i ð ÞÀ ð i Á iÞ i ¼ to 6 angular momentum integrals: ^ where q : ox ; oy ; roz is the gradient operator. r i ¼ð i i i Þ N When j? 0, the terms obviously vanish, the arc miqi q_ i C; distance tends to the Euclidean one, and the potential i 1 ^ ¼ ¼ becomes the standard Newtonian potential, in other words X where C = cwxew ex + cwyew ey + cwzew ez + cxyex ey + we recover the classical equations of motion, which we have ^ ^ ^ ^ cxzex ez + cyzey ez, with ew = (1, 0, 0, 0), ex = (0, 1, 0, 0), thus extended to spaces of constant curvature. The force acts ^ ^ ey = (0, 0, 1, 0), and ez = (0, 0, 0, 1) being the unit vectors in along geodesics: if two bodies are placed on the manifold 4 R ; whereas their wedge products form a basis of bivectors in with zero initial velocities, they will move along the common the corresponding exterior algebra. The real coefficients cwx, geodesic until they collide, the same as in the Euclidean case. cwy, cwz, cxy, cxz, cyz express the rotation of the system relative So everything makes sense from the physical point of view. to the planes given by their indices. Indeed, although it The next things to look for are the first integrals, since 3 makes sense in R to define rotation relative to an axis, it is they not only allow the reduction of the system, but also 4 better to think of rotations in R relative to an invariant (but have useful physical meaning. not pointwise fixed) plane. Unlike in Euclidean space, there are no integrals of the First Integrals centre of and linear momentum. This fact is not Before finding the first integrals, let’s simplify the problem. unexpected, given that these first integrals don’t exist for N- With the coordinate- and time-rescaling transformations body problems obtained by discretizing Einstein’s field

Ó 2013 Springer Science+Business Media New York, Volume 35, Number 3, 2013 27 Author's personal copy equations, as done by Levi-Civita [24], Einstein, Infeld, and Hoffmann [18], or Fock [19]. So, in general, there are no points like the centre of mass of a particle system in the Euclidean case, where forces cancel each other or make the centre move uniformly along a geodesic. Therefore the 2 2 dimension of the phase space q; q_ is 4N - 4 in S and H , ð Þ 3 3 since there are 4 first integrals, but 6N - 7 in S and H , because there are 7 first integrals.

Singularities These equations of motion open a new world. Any math- ematician would now want to find some solutions, hoping to understand how they behave. For given initial data, standard results of the theory of ordinary differential Figure 3. The initial positions of m1, m2, and m3 on the equations ensure the existence and uniqueness of an ana- geodesic z = 0. lytic solution, defined locally on some interval, say, [0, t+). Such a solution can be analytically extended to an interval corresponding to m1 and m2 at (0, 1) and m3 at (0, -1). + 0; tà , with 0 \ t \ tÃ. If tà , the solution is globally More precisely: ½defined,Þ i.e., the motion of the¼ bodies1 takes place forever. = But can it happen that tà is finite? (i) for M 8m, if the bodies are placed with zero initial To answer this question, we need to determine whether velocities close enough to the above collision-antipo- the equations of motion have singularities. A brief look dal configuration, they end up in a collision-antipodal reveals that they do. Whenever q D, where singularity at finite tÃ; 2 (ii) for M = 2m, if the bodies are placed, with zero initial 2 D 1 i\j N q qi qj 1 ; velocities, no matter how close to the collision-antip- ¼[   f jð Á Þ ¼ g odal configuration, they move away from it, never to at least a denominator cancels, so the equations fail to reach it; make sense. Consequently, for a solution q, the time tà (iii) for M = 4m, there exist initial positions, corresponding could be finite if limt t q t D. To understand what ! à ð Þ2 to zero initial velocities, for which the bodies reach the happens, notice first that D Dþ DÀ, where ¼ [ collision-antipodal configuration, but the solution remains analytic at t , so it does not encounter a Dþ 1 i\j N q qi qj 1 and à ¼[   f j Á ¼ g singularity there [15]. DÀ 1 i\j N q qi qj 1 : ¼[   f j Á ¼À g 2 3 Part of this highly unusual behaviour can be explained A close look shows that, in H or H , if q tends to an by the following result, which extends to 2 a theorem element of D, its limit must be an element of DÀ, which S 2 Painleve´ proved in 1896 for the classical 3-body problem. represents a collision between at least two bodies. In S or 3 The curved version below says that, to have expectable S , however, if q tends to an element of D, the solution could end up in a collision configuration, if the element is behaviour, we should stay away from collision-antipodal configurations [5]. in Dþ, an antipodal configuration, if the element is in DÀ, or a collision-antipodal configuration, which requires at least If q; q_ is an analytic solution of the curved 3-body problemð inÞ 2, defined on 0; t , with t a singularity, then 3 bodies and that the element be in the union Dþ DÀ. So S à à [ ½ Þ while in hyperbolic space there are no surprises, spherical 2 lim min qi qj 1 0: t t 1\i j\n space reveals something new. ! à  jð Á Þ À j¼ Further investigation shows that, for 2 bodies, collisions Conversely, assume that q; q_ is an analytic solution, on the sphere are ‘‘attractive,’’ whereas antipodal singu- ð Þ defined on 0; tà , that is bounded away from collision- larities are ‘‘repelling.’’ More precisely, if 2 bodies are antipodal configurations.½ Þ Then, if placed close to each other with zero initial velocities, they soon collide. But if 2 bodies are placed with zero initial lim min qi qj 1 0; t tà 1\i j\n j Á À j¼ velocities close to the opposite sides of a , they !  move fast away from each other along the common geo- tà is a singularity of the solution. desic, just to end up in a collision, too. But to understand what happens from the physical point Things become even more interesting with the collision- of view, we have to take a better look at the force function antipodal singularities. For simplicity, let us consider and its gradient. an isosceles problem on the equator z = 0, as shown in Figure 3. The bodies m1 and m2, which we keep symmetric A Cosmological Detour relative to the vertical axis, have equal mass M, whereas the Our intuition for gravitation follows the Newtonian view in body m3, which stays fixed at (x, y) = (0, -1), has mass Euclidean space: the larger the distance, the weaker the m. The study of this problem teaches us that anything can force. This model stays unchanged in the hyperbolic case. happen relative to a the collision-antipodal configuration But things are different on the sphere. Let’s start with N = 2

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2 on S and fix a body initially at the north pole. The space, are summarized in Table 1 [7]. The possibility of potential and the norm of its gradient are, respectively, such applications was a welcome bonus, but I had no desire to pursue it. had already answered m m q q U q 1 2ð 1 Á 2Þ ; the cosmological questions pertaining to this model. I ð Þ¼ 1 q q 2 1=2 ½ Àð 1 Á 2Þ Š therefore preferred to remain focused on studying the mathematical aspects of the curved N-body problem.

1=2 m2m2 q q q q 2 m2m2 q q q q 2 Relative Equilibria U q 1 2j 2 Àð 1 Á 2Þ 1j 1 2j 1 Àð 1 Á 2Þ 2j : 2 3 2 3 When entering a new world, it is always good to start from jr ð Þj¼ "#1 q1 q2 þ 1 q1 q2 ½ Àð Á Þ Š ½ Àð Á Þ Š simple things and make small, but solid steps forward. In other words, it’s better to understand a little than to mis- For various positions of the other body, U ranges from understand a lot. After taking a peek at singularities, the at collision to at the antipodal configuration, beingþ1 0 À1 first problem I posed to my collaborators was that of when the second body is on the equator. The norm of the finding relative equilibria (RE), i.e., orbits for which the gradient is at collision, and becomes smaller when the þ1 system behaves like a rigid body by maintaining equal second body lies farther away from collision in the north- mutual distances all along the motion. ern hemisphere; it takes a positive minimum value on the This is an active subject in the Euclidean case, where it has equator; and becomes larger the farther the second body is generated a lot of good mathematics. One of its questions from the north pole while in the southern hemisphere; the made Steven Smale’s list of open problems for the 21st century norm of the gradient becomes when the two bodies lie [35]. Saari’s conjecture, still an open problem in the general þ1 3 at antipodes. These remarks are also true in S . case, is also related to it [9, 13, 32]. But while this is an old topic The qualitative behaviour described above agrees with in Euclidean space, the tools used there don’t work in the the Newtonian gravitation in flat space only when the curved problem. Even its standard formulation is difficult to second body doesn’t leave the northern hemisphere. But in mimic in the new context, although it can be done [14]. a hypothetical spherical universe with trillions of objects The clue to how to start came from geometric ejected from a Big-Bang that took place at the north pole, mechanics, a research direction initiated by Ralph Abraham all the bodies would be now still far away from the equator, and Jerry Marsden, where a RE is defined as a solution since we know that, even if positive, the curvature would generated by a one-parameter subgroup of some Lie group be extremely small. When the expanding system approa- [1]. In this particular case, the Lie groups to look at are 2 3 2 3 ches the equator, many bodies get close to antipodal those that give the isometries of S ; S ; H , and H , i.e., 2;1 3;1 singularities, without coming close to collisions (due to the SO 3 ; SO 4 ; Lor R , and Lor R , respectively. Let us growing distance between any two bodies, as achieved by focusð Þ on theð Þ secondð andÞ the fourth,ð sinceÞ the other two are choosing the right initial conditions), so the potential respective subgroups of those. It is suitable in our case to energy becomes positive, like the kinetic energy. By the represent them in matrix form. 4 energy integral, the potential energy cannot grow beyond After fixing a suitable basis in R , the elements of SO(4), the value of the energy constant, which, when reached, which are orthogonal matrices of determinant 1, have the makes the kinetic energy zero and stops any motion. form PAP-1, where P SO 4 and The larger the initial velocities, the larger the energy con- 2 ð Þ cos h sin h 00 stant, but it is finite nevertheless, so the motion must stop at À some instant, to reverse from expansion to contraction. In a sin h cos h 00 A 0 1; highly populated spherical universe, the motion is contained ¼ 0 0 cos / sin / B À C in the northern hemisphere, away from the equator, never B 0 0 sin cos C B / / C able to cross into the southern hemisphere. This happens @ A only if all bodies are initially in the northern hemisphere, a with h; / R. After fixing a suitable basis in the Minkowski 3;21 3;1 restriction we don’t have to take into consideration for a space R , the elements of the Lorentz group Lor R , 3 3 -1 ð -Þ1 general dynamical study in S . When the motion takes place which leave H invariant, are of the form PBP or PCP , 3 3 3;1 in R or H , the Big-Bang could lead to a finite, eventually where P Lor R , 2 ð Þ collapsing, or infinite, eternally expanding universe, cos u sin u 00 depending on the initial velocities taken close to a singularity. À sin u cos u 00 These cosmological conclusions, which are in agree- B 0 1 and ment with those of general relativity, and do not require a ¼ 0 0 cosh s sinh s B C cosmological force, as Newtonian cosmology does in flat B 0 0 sinh s cosh s C B C @ 10 0 0 A Table 1. Cosmological behaviour in flat and curved classical universes 01 nn Geometry Volume Fate 0 1 C À 2 2 ; 3 ¼ 0 n 1 n =2 n =2 Elliptic, S Finite Eventual collapse B À2 2 C 3 B 0 n n =21n =2 C Euclidean, R Finite or infinite Eternal expansion or eventual collapse B À þ C Hyperbolic, 3 Finite or infinite Eternal expansion or eventual collapse @ A H with u; s; n R. 2

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For A, we call the rotations positive elliptic-elliptic if h, / = 0, and positive elliptic if h = 0 and / = 0 (or h = 0 and / = 0, a case we ignore since it’s the same as the previous); for B, negative elliptic if u = 0 and s = 0, negative hyperbolic if u = 0 and s = 0, and negative elliptic-hyperbolic if u, s = 0; for C, negative parabolic if n = 0. These possibilities lead to the natural definition of six types of RE, each corresponding to some type of rotation. It is easy to see that the negative parabolic RE cannot satisfy all integrals of the angular momentum, so such solutions don’t exist. But the other classes of RE seem to be rich, and the solutions they contain are often surprising [6, 8, 12, 14]. Among them are some orbits of the 6-body Figure 4. A 3-dimensional projection of a 4-dimensional 3 3 problem of equal in S for which 3 bodies are at the foliation of the sphere S into Clifford tori. vertices of an equilateral triangle inscribed in the

1 2 2 S : w; x; y; z y z 1; w x 0 ; 3 4 wx ¼fð Þj þ ¼ ¼ ¼ g The RE of S live on Clifford tori, which are tori of R : whereas the others are at the vertices of an equilateral tri- T2 w; x; y; z r2 2 1; 0 ; 2 ; 1 rq q h /\ p angle inscribed in the circle Syz, whose meaning is obvious ¼fð Þj þ ¼  g from the above. For one type of solution, a triangle rotates where w r cos h; x r sin h; y q cos /, and z q sin /. 1 1 ¼ ¼ ¼3 ¼ uniformly along Swx, whereas the other is fixed in Syz. For Unlike the standard torus of R , these surfaces are flat. 3 the other type of solution, both triangles rotate uniformly Moreover, they are subsets of S . Each Clifford torus 3 along their respective , with whatever angular velo- divides S into two solid tori, forming the boundary 3 cites we assign to each of them. This means that the orbits between them. The sphere S can also be foliated into 3 are not periodic, but quasiperiodic, a situation that cannot Clifford tori, as Figure 4 shows. The RE of H live instead 3 be encountered in R . on hyperbolic cylinders, How are such exotic orbits possible? The answer lies in 2 2 2 3 1 1 Crg w; x; y; z r g 1; 0 h\2p; n R ; the geometry of S . The circles Swx and Syz are special, ¼fð Þj À ¼À  2 g forming a Hopf link in a Hopf fibration, which is the where w rcosh; x rsinh; y gsinhn; z gcoshn. Unlike mapping the Clifford¼ tori, they¼ are surfaces¼ of positive¼ Gaussian cur- 3 3 2 vature. As in the case of the RE of , the qualitative : S S ; w; x; y; z S H ! Hð Þ 3 w2 x2 y2 z2; 2 wz xy ; 2 xz wy behaviour of each type of orbit in H can be described, but ¼ð þ À À ð þ Þ ð À ÞÞ we are not going into these details here. 3 2 that takes circles of S to points of S . In particular, Instead, I briefly discuss some simple orbits, which bear 1 1 H applies Swx to (1, 0, 0) and Syz to (-1, 0, 0). It can be proved the name of Lagrange, who discovered them in 1772 in the with the help of stereographic projection that these two Euclidean case. These RE are formed by equilateral trian- 2 circles are linked like adjacent rings in a chain. Moreover, if gles that rotate around the centre of mass. In R the masses 1 1 2 2 a Swx and b Syz, then d(a, b) = p/2, i.e., the arc dis- can have any values, but in S and H they must be equal, a tance2 between2 the two circles is constant. So no matter property that occurs because spheres and hyperbolic where two bodies lie (one on each of these circles), the spheres have fewer than flat space. magnitude of the attraction force between them is the To understand the importance of this remark, let us same. By suitably aligning the direction of the forces acting recall an old problem Gauss dealt with: the shape of between the two circles, as achieved with the help of two physical space. In the early 1820s, he went as far as to equilateral triangles, it is possible to obtain orbits like the measure the angles of the triangle formed by three moun- ones described above. tain peaks near Go¨ttingen (Inselberg, Brocken, and Hoher Many other interesting RE exist, such as one of the Hagen), to see whether their sum deviated from p radians 4-body problem in which the bodies rotate at the vertices of ([7, 8, 21, 30]) and thus prove space hyperbolic or elliptic. a regular tetrahedron, and of the 5-body problem where His experiment failed because he found no deviation the bodies are at the vertices of a uniformly rotating pen- beyond the inevitable measurement errors. Gauss’s method 3 tatope (4-simplex) in S . All solutions described so far are cannot provide an answer for cosmic triangles because we positive elliptic-elliptic RE. cannot reach distant stars to measure the required angles. For negative curvature, unusual orbits occur too. For But if we can mathematically prove the existence of instance, certain negative hyperbolic RE of the 3-body celestial orbits specific only to one of the hyperbolic, flat, or 3 problem move in H like airplanes flying in formation, all elliptic space, and none of the other two, then we might be on the same moving geodesic: the pilot in the middle sees able to decide on the shape of physical space by seeking always the other two planes left and right of him, as if such orbits in the night sky through astronomical obser- unmoved. Such orbits are not even quasiperiodic, and vations. We know, however, that Lagrangian orbits exist in don’t come close to any periodicity at all. our solar system, as for example those formed by the Sun,

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Jupiter, and any of the Greek or Trojan asteroids. In the the end, they gave me a new angle from which to look at light of the previous result on Lagrangian orbits, does this the problem, and soon I understood that I had been wrong. mean that, at least for distances of order of 10 AU, space is There is a classical result in Lie group theory (of which I Euclidean? was not aware before), which says that every element of a Such a conclusion would be premature, but we might semisimple compact Lie group lies in some maximal torus. not be too far from it. We would first have to make sure that SO(4) is such a Lie group, and its maximal tori are of the form 3 3 4 the equality of the masses must also occur in S and H , SO(2) 9 SO(2). By fixing a coordinate system in R , we also and that there are no quasiperiodic orbits of nonequal define two rotations, so we fix the underlying SO(2) 9 SO(2) masses in curved space that are close to the Lagrangian subgroup. And, indeed, in this naturally fixed subgroup, ones. We did not prove such results yet, and don’t even there is no element that could generate the Lagrangian RE. know whether they are true. But this is an incentive to But there is always another system of coordinates, which better understand the equations of the curved and flat N- fixes a different torus, where we can find such an element, as body problem. the theorem taught me. In other words, I had looked at the Lagrangian RE in the wrong coordinate system. It was like Rotopulsators watching the uniform motion of a point around a circle in the An interesting class of orbits in the Euclidean case is that of projection of this circle (an ellipse) on some inclined plane. homographic solutions. As their name suggests, they form Pursuing the due diligence helped me evade the trap. It was configurations that are self-similar during the motion. But not the first time. Traps, in various disguises, showed up and on spheres and hyperbolic spheres the only similar con- will likely show up again in this world where many things are figurations are the congruent ones, since angles change surprising and unexpected. when increasing or decreasing a given shape, even if the distances remain proportional. Of course, if viewed in the Stability background space, the Euclidean figures are similar, but we Trying to decide the curvature of physical space using the like to think in terms of the space’s natural geometry. idea described above would make little sense if the orbits I therefore needed a new name for orbits given by found mathematically to exist are not stable. Of course, the configurations that rotate and expand or shrink at the same stability of an orbit does not guarantee that it occurs in time. After some reflection, I came up with the term roto- nature, but instability makes certain that we won’t find it in pulsator, which captures the above properties without the universe. So studying the stability of orbits is an implying the geometrical similarity of the configuration. We important, though difficult, mathematical task. can think of an orbit that, say, only expands, as having a There are many kinds of stability for solutions of single pulse. The definitions were easy to obtain from RE, by ordinary differential equations. The most desirable is the allowing dilation and contraction and, therefore, a non- one defined by Lyapunov, which assures that nearby orbits constant angular velocity. So I was led to 5 classes of stay close for all time. But this property is rare, and the 3 3 rotopulsators: 2 in S and 3 in H , and I started studying more so in celestial mechanics, where what one usually them with my students Shima Kordlou and Brendan Thorn hopes for is orbital stability, which means that at least the [10, 17]. Then I also employed a young man, Sergiu Popa, orbits, but not necessarily the positions and velocities of the who—due to personal circumstances—is currently not bodies, remain close to each other when small perturba- enrolled in the academic system, but has high talent for tions occur. Most of the time, however, we are content to mathematics and does excellent research. We started prove even weaker properties, such as linear stability, developing this topic and found many new classes of orbits. which does not necessarily imply stability in nature, But instead of describing such solutions, I will reveal a although it gives hopes for it. trap I fell into, which shows why research in uncharted In Euclidean space we know that the Lagrangian orbits are territory must be advanced with care and restraint. In the Lyapunov stable if one of the masses is very small, but class of what I called positive elliptic-elliptic rotopulsators unstable if the masses are comparable. Therefore one would 3 (orbits with two rotations in S ), I checked to see whether expect that these orbits are unstable in curved space, since Lagrangian orbits of equal masses occur in the 3-body case. they occur only for equal masses. But this conclusion might They do, but the ones I found were very strange: the sides not even be true. In a recent paper, Regina Martı´nez and of the equilateral triangle are constant, so there is no Carles Simo´ showed that there are zones of linear stability for 2 expansion or contraction. In other words, they are RE. Still, the Lagrangian orbits on S [29]. As expected, that does not there is no element in the natural underlying subgroup happen when the equilateral triangle is small, since we get SO(2) 9 SO(2) of SO(4) that can generate this orbit. This close to the Euclidean case, but it occurs when the size of the orbit, however, passes at every instant through a contin- system is large enough. Similar things happen for other types uum of SO(2) 9 SO(2) elements. of orbits, such as the tetrahedral ones in the 4-body problem 2 Since researchers in geometric mechanics define RE as in S , as shown in [11]. orbits for which there is an element of some Lie group that can generate the orbit, I had apparently stumbled into an Perspectives example that showed their definition to be inadequate. And When I present this topic to seasoned researchers, they for a couple of weeks I thought that this was the case, so I often ask me whether I tried to answer this or that question told my friends Tudor Ratiu and James Montaldi about it. In about the equations of motion. Most of the time my answer

Ó 2013 Springer Science+Business Media New York, Volume 35, Number 3, 2013 31 Author's personal copy is no. There is so much to discover in this new world (and [6] F. Diacu, Polygonal homographic orbits of the curved 3-body the questions come naturally, by comparing what was done problem, Trans. Amer. Math. Soc. 364 (2012), 2783–2802. in the Euclidean case, which has been researched for more [7] F. Diacu, Relative equilibria of the 3-dimensional curved n-body than 3 centuries) that, with my few collaborators and stu- problem, Memoirs Amer. Math. Soc. (to appear). dents, I have only scratched the surface so far. [8] F. Diacu, Relative equilibria of the curved N-body problem, This problem also lays bridges between several bran- Atlantis Press, Paris, 2012. ches of mathematics: differential equations, dynamical [9] F. Diacu, T. Fujiwara, E. Pe´ rez-Chavela, and M. Santoprete, systems, geometric mechanics, non-Euclidean and differ- Saari’s homographic conjecture of the 3-body problem, Trans. ential geometry, the theory of polytopes, geometric Amer. Math. Soc. 360, 12 (2008), 6447–6473. topology, Lie groups and algebras, and stability theory [11], [10] F. Diacu and S. Kordlou, Rotopulsators of the curved N-body while also showing potential for applications. problem, arXiv:1210.4947. Obvious mathematical questions to ask are related to [11] F. Diacu, R. Martı´nez, E. Pe´ rez-Chavela, and C. Simo´ , On the regularizing collisions and better understanding the other stability of tetrahedral relative equilibria in the positively curved singularities that show up; discovering more non-equal 4-body problem, Physica D (to appear). mass orbits and determining their stability (spectral, linear, [12] F. Diacu and E. Pe´ rez-Chavela, Homographic solutions of the orbital, nonlinear); finding the bifurcations that occur when curved 3-body problem, J. Differential Equations 250 (2011), passing from negative to positive curvature through the Euclidean case, developing a 3D theory in terms of intrinsic 340–366. equations, which were written only in 2D so far; seeking [13] F. Diacu, E. Pe´ rez-Chavela, and M. Santoprete, Saari’s conjec- new RE and rotopulsators and understanding their prop- ture for the collinear N-body problem, Trans. Amer. Math. Soc. erties; solving the curved version of Saari’s conjecture and 357, 10 (2005), 4215–4223. the corresponding Smale problem, which seems much [14] F. Diacu, E. Pe´ rez-Chavela, and M. Santoprete, The N-body harder than in the Euclidean case; and understanding problem in spaces of constant curvature. Part I: Relative whether chaotic behaviour shows up. Approaching these equilibria, J. Nonlinear Sci. 22, 2 (2012), 247–266. questions will generate other questions, as happens in [15] F. Diacu, E. Pe´ rez-Chavela, and M. Santoprete, The N-body every branch of mathematics. problem in spaces of constant curvature. Part II: Singularities, J. A recent attempt also shows the way into the field of Nonlinear Sci. 22, 2 (2012), 267–275. partial differential equations: I started working with my [16] F. Diacu, E. Pe´ rez-Chavela, and J. Guadalupe Reyes Victoria, An colleague Slim Ibrahim and our student Crystal Lind on intrinsic approach in the curved N-body problem. The negative understanding the Vlasov-Poisson equations for stellar curvature case, J. Differential Equations 252 (2012), 4529–4562. dynamics in spaces of constant curvature, which provide [17] F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body the natural extension of the curved N-body problem to problem, arXiv:1302.5352. kinetic theory. This question is independent of everything [18] A. Einstein, L. Infeld, and B. Hoffmann, The gravitational equations mentioned in the previous paragraph and it will likely have and the problem of motion, Ann. of Math. 39, 1 (1938), 65–100. a life of its own. [19] V. A. Fock, Sur le mouvement des masses finie d’apre` s la the´ orie I have illustrated here both the risks and the rewards de gravitation einsteinienne, J. Phys. Acad. Sci. USSR 1 (1939), encountered when stepping into a new world. Thoughts of 81–116. this kind belong to our research lives, but they never make [20] I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice- it into journal papers. I have also tried to convey the idea Hall, Englewood Cliffs, NJ, 1963. that mathematics is not, as many people think, just about [21] G. Goe, Comments on Miller’s ‘‘The myth of Gauss’s experiment proving a famous conjecture, but it is first of all about finding new viable paths that keep our curiosity alive. 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[28] N. I. Lobachevsky, The new with full [32] D. Saari, Collisions, Rings, and Other Newtonian N-body theory of parallels [in Russian], 1835–1838, In Collected Works, Problems, American Mathematical Society, Regional Conference V. 2, GITTL, Moscow, 1949, p. 159. Series in Mathematics, No. 104, Providence, RI, 2005. [29] R. Martı´nez and C. Simo´ , On the stability of the Lagrangian [33] E. Schering, Die Schwerkraft im Gaussischen Raume, Nachr. 2 homographic solutions in a curved three-body problem on S , Ko¨ nigl. Gesell. Wiss. Go¨ ttingen 13 July, 15 (1870), 311–321. Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 1157–1175. [34] A.V. Shchepetilov, Nonintegrability of the two-body problem in [30] A. I. Miller, The myth of Gauss’s experiment on the Euclidean constant curvature spaces, J. Phys. A: Math. Gen. V. 39 (2006), nature of physical space, Isis 63, 3 (1972), 345–348. 5787–5806; corrected version at math.DS/0601382. [31] E. Pe´ rez-Chavela and J.G. Reyes Victoria, An intrinsic approach [35] S. Smale, Mathematical problems for the next century, Mathe- in the curved N-body problem. The positive curvature case, matics: frontiers and perspectives, Amer. Math. Society, Trans. Amer. Math. Soc. 364, 7 (2012), 3805–3827. Providence, RI, 1999, pp. 271–294.

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