By the spring of 2014 I had largely given up on the three-body problem. Out of ideas, I began programming on my laptop to generate and search through approximate solutions. These attempts would never solve my problem outright, but they might garner evidence toward an answer. My lack of programming expertise and resulting impatience slowed the process, making it an unpleasant experience for a pencil-and-paper mathematician like myself. I sought out my old friend Carles Simó, a professor at the University of Barcelona, to convince him to aid me in my clunky search. That fall I traveled to Spain to meet with Simó, who had a reputation as one of the most inventive and careful numerical analysts working in celestial mechanics. He is also a direct man who does not waste time or mince words. My first afternoon in his office, after I had explained my question, he looked at me with piercing eyes and asked, “Richard, why do you care?” The answer goes back to the origins of the three-body problem. Isaac Newton originally posed and solved the two-body problem (see Figure 1) when he published his Principia in 1687. He asked: “How will two masses move in space if the only force on them is their mutual gravitational attraction?” Newton framed the question as a problem of solving a system of differential equations—equations that dictate an object’s future motion from its present position and velocity. He completely solved his equations for two bodies. The solutions, also called orbits, have each object moving on a conic—a circle, ellipse, parabola or hyperbola. In finding all the possible orbits, Newton derived Johannes Kepler’s laws of planetary motion, empirical laws Kepler published in 1609 that synthesized decades of astronomical observations by his late employer, Tycho Brahe. Kepler’s first law says that each planet (or comet) moves on a conic with the sun as its focus. In Newton’s solutions, however, the two bodies—the sun and a planet—move on two separate conics. These conics share one focus, which is the center of mass of the two bodies. The sun is more massive than any planet, so much so that the center of the mass of the sun-planet system is inside the sun itself, very close to the sun’s center of mass, with the sun’s center of mass barely wobbling about the common center on a tiny elliptical path. In place of the two masses, put three, and you have the three-body problem. Like its predecessor, its orbits are solutions to a system of differential equations. Unlike its predecessor, however, it is difficult to impossible to find explicit formulas for the orbits. To this day, despite modern computers and centuries of work by some of the best physicists and mathematicians, we only have explicit formulas for five families of orbits, three found by Leonhard Euler (in 1767) and two by Joseph-Louis Lagrange (in 1772) (see Figure 2). In 1890 Henri Poincaré discovered chaotic dynamics within the three-body problem, a finding that implies we can never know all the solutions to the problem at a level of detail remotely approaching Newton’s complete solution to the two-body problem. Yet through a process called numerical integration, done efficiently on a computer, we can nonetheless generate finite segments of approximate orbits, a process essential to the planning of space missions. By extending the run-time of the computer, we can make the approximations as accurate as we want. Eclipses Simó’s words had knocked the breath out of me. “Of course, I care,” I thought. “I have been working on this problem for nearly two decades!” In fact, I had been focusing on a particular question within the problem that interested me: Is every periodic eclipse sequence the eclipse sequence of some periodic solution to the planar three-body problem? Let me explain. Imagine three bodies—think of them as stars or planets—moving about on a plane, pulling at one another with gravity. Number the bodies one, two and three. From time to time all three will align in a single, straight line (see Figure 3). Think of these moments as eclipses. (Technically, this “eclipse” is called a syzygy, an unbeatable word to use in hangman.) As time passes, record each eclipse as it occurs, labeling it one, two or three, for whichever star is in the middle. In this way, we get a list of ones, twos and threes called the eclipse sequence. For example, in a simplified version of our sun-Earth-moon system, the moon (which we will label body “3”) makes a circle around Earth (body “2”) every month, while Earth makes a circle around the sun (body “1”) once a year. This movement is repetitive, so it will give us a periodic eclipse sequence. Specifically: 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3. There is no 1 in the sequence because the sun never lands between Earth and the moon. In one year, the list is 24 numbers long, with a 2, 3 for each of the 12 months of the year. There is no reason that the eclipse sequence of a solution must repeat itself. It might go on forever with no discernible pattern. If, however, the solution exactly repeats itself after some period of time, like the Earth-moon-sun system after a year, then the sequence repeats: the same 24 numbers of the Earth-moon-sun system replay each year. So, returning to my question: Is every periodic eclipse sequence the eclipse sequence of some periodic solution to the planar three-body problem? I suspected the answer was yes, but I could not prove it. Holey Objects To justify the importance of my question, I reminded Simó of a basic fact tying together three branches of mathematics: topology, sometimes called rubber-sheet geometry; Riemannian geometry, the study of curved surfaces; and dynamics, the study of how things move. Imagine a bug walking along a curved surface shaped like the “wormhole surface,” also called a catenoid (see Figure 4). The bug’s job is to find the shortest circuit going once around the hole. As far as topology is concerned, the wormhole surface (see Figure 5) is the same as the x-y-plane with a single hole punctured in it. Indeed, imagine a hole punctured into a flexible rubber sheet. By pushing the hole downward and stretching it outward, you can make the wormhole surface. If the hole has been sufficiently flared outward, then not only does this shortest circuit exist, but it satisfies a differential equation very much like the three-body equations. In this way, our bug has found a periodic solution to an interesting differential equation. In the three-body problem, the role of the wormhole surface is played by something called configuration space—a space whose points encode the locations of all three bodies simultaneously, so that a curve in configuration space specifies the motions of each of the three bodies. By insisting that our bodies do not collide with one another, we pierce holes in this configuration space. As we will see, as far as topology, or rubber-sheet geometry, is concerned, the resulting collision-free configuration space is the same as an x-y-plane with two holes punctured in it (see Figure 6). We will label the holes as “12,” meaning bodies 1 and 2 have collided, and “23,” meaning that 2 and 3 have collided, and place the holes on the x-axis. We’ll also place a third hole at infinity and label it “13” to represent bodies 1 and 3 colliding. These holes break the x-axis into three segments labeled 1, 2 and 3. A curve in this twice-punctured plane represents a motion of all three bodies—which is to say, a potential solution to the three-body problem. When the curve cuts across segment 1, it means an eclipse of type 1 has occurred and likewise for cutting across segment 2 or 3. In this way, an eclipse sequence represents a way of winding around our collision holes. Now, our bug was trying to minimize the length of its path as it circled once around the wormhole. To get the correct analogy between the bug’s problem and the three-body problem, we must replace the length of a path by a quantity called the action of a path. (The action is a kind of average of the instantaneous kinetic energy minus the potential energy of the motion represented by the path.) A centuries-old theorem from mechanics states that any curve in configuration space that minimizes the action must be a solution to Newton’s three-body problem. We can thus try to solve our eclipse sequence problem by searching, among all closed paths that produce a fixed eclipse sequence, for those closed paths that minimize the action. This strategy—seeking to minimize the action in configuration space for loops having a particular eclipse sequence—had preoccupied me for most of 17 years and led to many nice results. For instance, in 2000 Alain Chenciner of Paris Diderot University and I rediscovered what seems to be the first known periodic solution to the three-body problem with zero angular momentum. It was a figure-eight-shaped solution (see Figure 7) first found by Cris Moore of the Santa Fe Institute in 1993. In this case, three equal masses chase one another around a figure-eight shape on the plane. Its eclipse sequence is 123123, repeating forever. Our work popularized the figure eight and gave it a rigorous existence proof. It also led to an explosion of discoveries of many new orbits for the equal-mass N-body problem, orbits christened “choreographies” by Simó (see Figure 8), who discovered hundreds of these new families of orbits. Our figure-eight orbit even made it into the best-selling Chinese science-fiction novel by Liu Cixin, whose English translation was entitled The Three-Body Problem. The morning after I shared my ponderings with Simó, he said something that affected me deeply. “Richard, if what you think about your question is true, then there must be a dynamical mechanism.” In other words, if I was right that the answer to my question was yes, then there must be something about how these bodies moved that made it so. Those few words made me question my convictions and led me to abandon my 17-year-long attempt to answer my question by minimizing the action of paths. What dynamical mechanisms in this problem did I even understand? I wondered. I could think of two, only one of which held out hope. This mechanism, related to the chaos discovered by Poincaré, led me to reflect on old work of a recent collaborator of mine, Rick Moeckel of the University of Minnesota. In the 1980s he had shown how curves called hyperbolic tangles, born from triple collisions in the three-body problem, can lead to astounding results. As I reread his old papers, it seemed to me that Moeckel had the key to my problem. I got in touch with him, and within a few days Moeckel and I had answered my question! Well, almost. We had answered a question infinitely close. The Shape Sphere Understanding Moeckel’s dynamical mechanism, in conjunction with the relationship between the three-body configuration space and the plane with two holes described above, requires thinking about an object called the shape sphere. As the three bodies move around in the plane, at each instant they form the three vertices of a triangle. Instead of keeping track of the position of each vertex, let us keep track of only the overall shape of the triangle. The result is a curve on the shape sphere, a sphere whose points represent “shapes” of triangles. What is a “shape”? Two figures in the plane have the same shape if we can change one figure into the other by translating, rotating or scaling it (see Figure 9). The operation of passing from the usual three-body configuration space—which is to say, from the knowledge of the locations of all three vertices of a triangle—to a point in the shape sphere, is a process of forgetting—forgetting the size of the triangle, the location of its center of mass, and the orientation of the triangle in the plane. That the shape sphere is two-dimensional is easy to understand from high school geometry: we know the shape of a triangle if we know all three of its angles, but because the sum of the three angles is always 180 degrees, we really only need two of the three angles—hence, two numbers are sufficient to describe the shape of a triangle. That the shape sphere is actually a sphere is harder to understand and requires that we allow triangles to degenerate, which is to say, we allow “triangles” consisting of three vertices that all lie on the same line to be called triangles. These so-called degenerate zero-area triangles form the equator of the shape sphere: they are the eclipses! The area of a triangle, divided by its size (r) squared, is its distance to the equator. The north and south poles of the sphere represent those triangles of maximum possible area and are the two equilateral triangle shapes. But why are there two equilateral shapes? These two equilateral triangle shapes differ by the cyclic order of their vertices (see Figure 9). There is no way to turn one of these equilateral triangles into the other by a rotation, translation or scaling of the plane: they represent different shapes. Yet the operation of reflecting about a line (any line) in the plane will turn one equilateral triangle shape into the other one. This reflection operation acts on all triangles, and so on the shape sphere itself, where it acts by reflection about the equator, keeping the points of the equator (degenerate triangles) fixed while interchanging the north and south hemispheres. Included among the degenerate triangles are the binary collisions: those “triangles” for which two of the three vertices lie on top of each other. There are exactly three of these binary collision triangles, labeled “12,” “23” and “13,” according to which two vertices lie on top of each other. I can now explain how the shape sphere shows us that the three-body configuration space is topologically the same as the usual x-y-plane minus two points. We have to know that the sphere minus a single point is topologically the same object as the usual x-y-plane (see Figure 10). One way to see this fact about the sphere is to use stereographic projection, which maps the sphere with a single point removed (the “light source”) onto the usual x-y-plane. As a point on the sphere tends toward the light source, its image point on the x-y-plane moves out to infinity, so we can also say that the plane with a point at infinity added is topologically equal to the sphere. Take the light source to be the 13 binary collision point of the shape sphere, so that the point at infinity of the x-y-plane corresponds to the 13 collision point. Orient the sphere so that its equatorial plane intersects the x-axis of the x-y-plane. Then stereographic projection maps the equator of degenerate triangles to the x-axis of the plane and the other two binary collision points get mapped to two points on this x-axis. In this way, we arrive at exactly the picture described earlier. The three binary collision points form three special points on the shape sphere. Besides these three there are additional special points on the shape sphere called central configurations. These five central configurations correspond to the five families of solutions discovered by Euler and Lagrange. Their solutions are the only three-body solutions for which the shape of the triangle does not change as the triangle evolves! In the Lagrange solutions, the triangle remains equilateral at each instant; there are two Lagrange configurations, as we have seen, and they form the north and south poles of the shape sphere (see Figure 11). We label them “Lagrange point 4” and “Lagrange point 5.” The remaining three central configurations are the Euler configurations, labeled “Euler point 1,” “Euler point 2” and “Euler point 3.” They are collinear (all in a line), degenerate configurations, so they lie on the equator of the shape sphere. They are positioned on the equator between the three binary collision points (see Figure 12). (Their spacing along the equator depends on the mass ratios between the three masses of the bodies.) Euler point 1, for example, lies on the equatorial arc marked 1, so is a collinear shape in which body 1 lies between bodies 2 and 3. (Often all five central configuration points are called Lagrange points, with the Euler points labeled “L1,” “L2” and “L3.”) One can understand the central configuration solutions by dropping three bodies, by which I mean, by letting the three bodies go from rest, with no initial velocity. Typically when one does this, all kinds of crazy things will happen: close binary collisions, wild dances and perhaps the escape of one body to infinity. But if one drops the three bodies when they are arranged in one of the five central configuration shapes, then the triangle they form simply shrinks to a point, remaining in precisely the same shape as it started, with the three masses uniformly pulling on one another until the solution ends in a simultaneous triple collision. The five roads to triple collision Triple collision is an essential singularity within the three-body problem, something like a big bang at the center of the problem, and it is the source of much of its chaos and difficulty. In the early 1900s Finnish mathematician Karl Sundman proved that the five central configurations, as represented by the dropped solutions just described, are the only roads to triple collision. What this means is that any solution that ends in triple collision must approach it in a manner very close to one of these five dropped central configuration solutions, and as it gets closer and closer to triple collision, the shape of the solution must approach one of the five central configuration shapes. Sundman’s work was a complicated feat of algebra and analysis. Then, in the year I graduated from high school (completely oblivious to the three-body problem), American mathematician Richard McGehee invented his so-called blow-up method, which allowed us to understand Sundman’s work pictorially and to study dynamics near triple collision in much greater detail. Let r denote the distance to triple collision—a measure of the overall size of a triangle. As r approaches zero, Newton’s equations become very badly behaved, with many terms going to infinity. McGehee found a change of configuration space variables and of time that slows down the rate of approach to triple collision and turns the triple collision point, which is r = 0, into an entire collection of points: the collision manifold. Surprise! The collision manifold is essentially the shape sphere. McGehee’s method extended Newton’s equations, originally only valid for r greater than zero, to a system of differential equations that makes sense when r = 0. Newton’s equations have no equilibrium points, meaning there are no configurations of the three bodies that stand still: three stars, all attracting one another, cannot just sit there in space without moving. But when Newton’s equations are extended to the collision manifold, equilibrium points appear. There are exactly 10 of them, a pair for each of the five central configuration points on the shape sphere. One element of a pair represents the end result of the corresponding dropped central configuration in its approach to triple collision. Newton’s equations stay the same even if we run time backward, so we can run any solution in reverse and get another solution. When we run a dropped central configuration solution backward, we get a solution that explodes out of triple collision, reaching its maximum size at the dropped configuration. The other element of the pair represents the initial starting point of this “exploding” solution. Together these two central configuration solutions—collision and ejection—fit smoothly and form a single ejection-collision solution that leaves the ejection equilibrium point at r = 0, enters into the r greater than zero region where it achieves a maximum size, and then shrinks back to end up on the triple collision manifold at the collision equilibrium point there. This complete solution connects one element of an equilibrium pair to the other. By creating these equilibrium points associated with central configurations, buried deep inside the three-body problem, McGehee gave Moeckel a key that enabled him to apply recently established results from modern dynamical systems—results unavailable to Newton, Lagrange or Sundman—to make some interesting headway on the three-body problem. Moeckel’s walk In Moeckel’s papers I saw a picture of a graph with five vertices labeled by the central configurations and joined together by edges (see Figure 13). A walk on a graph is a possible circuit through its vertices, traveling the edges from vertex to vertex. Moeckel proved that any possible walk you can take on his graph corresponds to a solution to the three-body problem that comes close for some time to the central configuration solution labeled by the corresponding vertex. For example, the walk E1 L4 E2 L5 corresponds to a solution very close to the Euler ejection-collision solution associated with the Euler point 1, then comes close to triple collision almost along the Lagrange L4 central configuration solution, but before total triple collision is achieved the three bodies shoot out along one of the five “roads” very near to the Euler point 2 central configuration solution. Then, finally, as this Euler solution collapses back toward triple collision, the solution spins out into a Lagrange L5 equilateral shape. Moreover, if we repeat this same walk, making it periodic, the solution following it will be periodic. Soon after Simó told me there had to be a dynamical mechanism, I realized that Moeckel’s graph embedded into the shape sphere (see Figure 14). The important thing about this embedded graph is that it carries all of the topology of the sphere with its three binary collision holes. Indeed, we can deform the thrice-punctured sphere onto the graph and in so doing turn any loop in the punctured sphere to a walk on the graph. To see this deformation, imagine the sphere as the surface of a balloon. Make three pin pricks in it, one at each binary collision hole. The balloon is made of very flexible material, so we can stretch out our three pinpricks, enlarging them until the edges of the three holes almost touch each other and the remaining material forms a ribbon hugging close to the embedded graph. In the process of making this deformation, any closed loop in the thrice-punctured sphere gets deformed into a closed loop in this ribbon structure and, from there, to a walk on Moeckel’s embedded graph. To turn this picture into a theorem about solutions, I needed to prove that if I project the solutions guaranteed by Moeckel’s theorem onto the shape sphere, then they never stray far from this embedded graph. If they did, they could wind around the binary collisions or even hit one, killing or adding some topologically significant loops and so changing the eclipse sequence. I e-mailed Moeckel to ask for help. He wrote back, “You mean you’re going to force me to read papers I wrote over 20 years ago?” Nevertheless, he dove back into his old research and proved that the projections of the solutions he had encoded symbolically all those years ago never did stray far from the embedded graph. My question was answered—almost. To make his proof work, Moeckel needed a tiny bit of angular momentum. (Angular momentum, in this context, is a measure of the total amount of “spin” of a system and is constant for each solution.) But for those 17 years before my conversation with Simó I had insisted on solutions having zero angular momentum. This insistence arose because solutions that minimize action among all curves having a given eclipse sequence must have zero angular momentum. On the other hand, Moeckel needed a small bit of angular momentum to get solutions traveling along the edges of his graph. The symbol for a tiny positive quantity in mathematical analysis is an epsilon. We needed an epsilon of angular momentum. There was another catch to Moeckel’s results: his solutions, when they cross the equator of the shape sphere near the Euler points E1, E2 and E3, will oscillate back and forth there across the equator before traveling up to the north or south pole as they go in near triple collision along the corresponding Lagrange road, L4 or L5. To account for these oscillations, take a positive integer N and call an eclipse sequence “N-long” if every time a number occurs in the sequence it occurs at least N times in a row. For example, the sequence 1112222333332222 is 3-long, but it is not 4-long, because there are only three 1s in a row. Here, finally, is our main theorem: Consider the three-body problem with small nonzero angular momentum epsilon and masses within a large open range. Then there is a large positive integer N with the following significance. If we choose any eclipse sequence whatsoever—which is N-long—then there is a corresponding solution to our three-body problem having precisely this eclipse sequence. If that sequence is made to be periodic, then so is the solution realizing it. What about my original question? There was no large N mentioned there. I had asked about every eclipse sequence. But I did not tell you my real question. What I really wanted to know was whether or not I could realize any “topological type” of periodic curve, not any eclipse sequence. I was using the eclipse sequence as a convenient shorthand or way of encoding topological type, which is to say as a way of encoding the winding pattern of the loop around the three binary collision holes. The eclipse sequence representation of the topological type of a closed curve has redundancies: many different eclipse sequences encode the same topological type of curve. Consider, for example, the topological type “go once around the hole made by excluding the binary collision 23.” The eclipse sequence 23 represents this topological type. But so do the eclipse sequences 2223, 222223 and 2333. Whenever we have two consecutive crossings of the arc 2, we can cancel them by straightening out the meanders, making the curve during that part of it stay in one hemisphere or the other without crossing the equator (see Figure 15). Indeed, we can cancel any consecutive pair of the same number that occurs in an eclipse sequence without changing the topological type of closed curve represented by the sequence. To use our main theorem to answer my real question, note that by deleting consecutive pairs I can ensure that the eclipse sequence that encodes a given topological type never has two consecutive numbers of the same type in it: no “11” or “22” or “33.” Call such a sequence an admissible sequence. Now, take any admissible sequence, for example, 123232. Allow me to use exponential notation in writing down eclipse sequences, so, for example, 13 = 111. Choose an odd integer n at least as big as the number N of our main theorem. Replace the admissible sequence by the longer sequence 1n 2n 3n 2n 3n 2n and continue it periodically. This longer sequence represents the same originally chosen topological type because n is odd. Our theorem says that this longer sequence is realized by a periodic solution. This periodic solution represents our original topological type 123232. What’s Next? We still have much left to do. When I originally posed my question almost 20 years ago, I only wanted solutions having zero angular momentum. But evidence is mounting that the answer to my question in the case of zero angular momentum is “no.” We have some evidence that even the simplest nonempty periodic sequence 23 is never realized by a periodic solution to the equal-mass, zero angular momentum three-body problem. Our main question as posed here, even for angular momentum epsilon, remains open because our theorem allowed us to realize only sequences that are N-long for some large N. We have no clue, for example, how to realize admissible sequences, that is, sequences with no consecutive numbers of the same type. At the end of the day, we may be no closer to “solving” the three-body problem in the traditional sense, but we have learned quite a lot. And we will keep at it—this problem will continue bearing fruit for those of us who are drawn to it. It turns out that new insights are still possible from one of the classic quandaries in mathematical history.
These attempts would never solve my problem outright, but they might garner evidence toward an answer. My lack of programming expertise and resulting impatience slowed the process, making it an unpleasant experience for a pencil-and-paper mathematician like myself. I sought out my old friend Carles Simó, a professor at the University of Barcelona, to convince him to aid me in my clunky search.
That fall I traveled to Spain to meet with Simó, who had a reputation as one of the most inventive and careful numerical analysts working in celestial mechanics. He is also a direct man who does not waste time or mince words. My first afternoon in his office, after I had explained my question, he looked at me with piercing eyes and asked, “Richard, why do you care?”
The answer goes back to the origins of the three-body problem. Isaac Newton originally posed and solved the two-body problem (see Figure 1) when he published his Principia in 1687. He asked: “How will two masses move in space if the only force on them is their mutual gravitational attraction?” Newton framed the question as a problem of solving a system of differential equations—equations that dictate an object’s future motion from its present position and velocity. He completely solved his equations for two bodies. The solutions, also called orbits, have each object moving on a conic—a circle, ellipse, parabola or hyperbola. In finding all the possible orbits, Newton derived Johannes Kepler’s laws of planetary motion, empirical laws Kepler published in 1609 that synthesized decades of astronomical observations by his late employer, Tycho Brahe. Kepler’s first law says that each planet (or comet) moves on a conic with the sun as its focus. In Newton’s solutions, however, the two bodies—the sun and a planet—move on two separate conics. These conics share one focus, which is the center of mass of the two bodies. The sun is more massive than any planet, so much so that the center of the mass of the sun-planet system is inside the sun itself, very close to the sun’s center of mass, with the sun’s center of mass barely wobbling about the common center on a tiny elliptical path.
In place of the two masses, put three, and you have the three-body problem. Like its predecessor, its orbits are solutions to a system of differential equations. Unlike its predecessor, however, it is difficult to impossible to find explicit formulas for the orbits. To this day, despite modern computers and centuries of work by some of the best physicists and mathematicians, we only have explicit formulas for five families of orbits, three found by Leonhard Euler (in 1767) and two by Joseph-Louis Lagrange (in 1772) (see Figure 2). In 1890 Henri Poincaré discovered chaotic dynamics within the three-body problem, a finding that implies we can never know all the solutions to the problem at a level of detail remotely approaching Newton’s complete solution to the two-body problem. Yet through a process called numerical integration, done efficiently on a computer, we can nonetheless generate finite segments of approximate orbits, a process essential to the planning of space missions. By extending the run-time of the computer, we can make the approximations as accurate as we want.
Eclipses
Simó’s words had knocked the breath out of me. “Of course, I care,” I thought. “I have been working on this problem for nearly two decades!” In fact, I had been focusing on a particular question within the problem that interested me:
Is every periodic eclipse sequence the eclipse sequence of some periodic solution to the planar three-body problem?
Let me explain. Imagine three bodies—think of them as stars or planets—moving about on a plane, pulling at one another with gravity. Number the bodies one, two and three. From time to time all three will align in a single, straight line (see Figure 3). Think of these moments as eclipses. (Technically, this “eclipse” is called a syzygy, an unbeatable word to use in hangman.) As time passes, record each eclipse as it occurs, labeling it one, two or three, for whichever star is in the middle. In this way, we get a list of ones, twos and threes called the eclipse sequence.
For example, in a simplified version of our sun-Earth-moon system, the moon (which we will label body “3”) makes a circle around Earth (body “2”) every month, while Earth makes a circle around the sun (body “1”) once a year. This movement is repetitive, so it will give us a periodic eclipse sequence. Specifically: 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3. There is no 1 in the sequence because the sun never lands between Earth and the moon. In one year, the list is 24 numbers long, with a 2, 3 for each of the 12 months of the year.
There is no reason that the eclipse sequence of a solution must repeat itself. It might go on forever with no discernible pattern. If, however, the solution exactly repeats itself after some period of time, like the Earth-moon-sun system after a year, then the sequence repeats: the same 24 numbers of the Earth-moon-sun system replay each year. So, returning to my question: Is every periodic eclipse sequence the eclipse sequence of some periodic solution to the planar three-body problem? I suspected the answer was yes, but I could not prove it.
Holey Objects
To justify the importance of my question, I reminded Simó of a basic fact tying together three branches of mathematics: topology, sometimes called rubber-sheet geometry; Riemannian geometry, the study of curved surfaces; and dynamics, the study of how things move. Imagine a bug walking along a curved surface shaped like the “wormhole surface,” also called a catenoid (see Figure 4). The bug’s job is to find the shortest circuit going once around the hole. As far as topology is concerned, the wormhole surface (see Figure 5) is the same as the x-y-plane with a single hole punctured in it. Indeed, imagine a hole punctured into a flexible rubber sheet. By pushing the hole downward and stretching it outward, you can make the wormhole surface. If the hole has been sufficiently flared outward, then not only does this shortest circuit exist, but it satisfies a differential equation very much like the three-body equations. In this way, our bug has found a periodic solution to an interesting differential equation.
In the three-body problem, the role of the wormhole surface is played by something called configuration space—a space whose points encode the locations of all three bodies simultaneously, so that a curve in configuration space specifies the motions of each of the three bodies. By insisting that our bodies do not collide with one another, we pierce holes in this configuration space. As we will see, as far as topology, or rubber-sheet geometry, is concerned, the resulting collision-free configuration space is the same as an x-y-plane with two holes punctured in it (see Figure 6). We will label the holes as “12,” meaning bodies 1 and 2 have collided, and “23,” meaning that 2 and 3 have collided, and place the holes on the x-axis. We’ll also place a third hole at infinity and label it “13” to represent bodies 1 and 3 colliding. These holes break the x-axis into three segments labeled 1, 2 and 3. A curve in this twice-punctured plane represents a motion of all three bodies—which is to say, a potential solution to the three-body problem. When the curve cuts across segment 1, it means an eclipse of type 1 has occurred and likewise for cutting across segment 2 or 3. In this way, an eclipse sequence represents a way of winding around our collision holes.
Now, our bug was trying to minimize the length of its path as it circled once around the wormhole. To get the correct analogy between the bug’s problem and the three-body problem, we must replace the length of a path by a quantity called the action of a path. (The action is a kind of average of the instantaneous kinetic energy minus the potential energy of the motion represented by the path.) A centuries-old theorem from mechanics states that any curve in configuration space that minimizes the action must be a solution to Newton’s three-body problem. We can thus try to solve our eclipse sequence problem by searching, among all closed paths that produce a fixed eclipse sequence, for those closed paths that minimize the action.
This strategy—seeking to minimize the action in configuration space for loops having a particular eclipse sequence—had preoccupied me for most of 17 years and led to many nice results. For instance, in 2000 Alain Chenciner of Paris Diderot University and I rediscovered what seems to be the first known periodic solution to the three-body problem with zero angular momentum. It was a figure-eight-shaped solution (see Figure 7) first found by Cris Moore of the Santa Fe Institute in 1993. In this case, three equal masses chase one another around a figure-eight shape on the plane. Its eclipse sequence is 123123, repeating forever. Our work popularized the figure eight and gave it a rigorous existence proof. It also led to an explosion of discoveries of many new orbits for the equal-mass N-body problem, orbits christened “choreographies” by Simó (see Figure 8), who discovered hundreds of these new families of orbits. Our figure-eight orbit even made it into the best-selling Chinese science-fiction novel by Liu Cixin, whose English translation was entitled The Three-Body Problem.
The morning after I shared my ponderings with Simó, he said something that affected me deeply. “Richard, if what you think about your question is true, then there must be a dynamical mechanism.” In other words, if I was right that the answer to my question was yes, then there must be something about how these bodies moved that made it so.
Those few words made me question my convictions and led me to abandon my 17-year-long attempt to answer my question by minimizing the action of paths. What dynamical mechanisms in this problem did I even understand? I wondered. I could think of two, only one of which held out hope. This mechanism, related to the chaos discovered by Poincaré, led me to reflect on old work of a recent collaborator of mine, Rick Moeckel of the University of Minnesota. In the 1980s he had shown how curves called hyperbolic tangles, born from triple collisions in the three-body problem, can lead to astounding results. As I reread his old papers, it seemed to me that Moeckel had the key to my problem. I got in touch with him, and within a few days Moeckel and I had answered my question! Well, almost. We had answered a question infinitely close.
The Shape Sphere
Understanding Moeckel’s dynamical mechanism, in conjunction with the relationship between the three-body configuration space and the plane with two holes described above, requires thinking about an object called the shape sphere. As the three bodies move around in the plane, at each instant they form the three vertices of a triangle. Instead of keeping track of the position of each vertex, let us keep track of only the overall shape of the triangle. The result is a curve on the shape sphere, a sphere whose points represent “shapes” of triangles.
What is a “shape”? Two figures in the plane have the same shape if we can change one figure into the other by translating, rotating or scaling it (see Figure 9). The operation of passing from the usual three-body configuration space—which is to say, from the knowledge of the locations of all three vertices of a triangle—to a point in the shape sphere, is a process of forgetting—forgetting the size of the triangle, the location of its center of mass, and the orientation of the triangle in the plane. That the shape sphere is two-dimensional is easy to understand from high school geometry: we know the shape of a triangle if we know all three of its angles, but because the sum of the three angles is always 180 degrees, we really only need two of the three angles—hence, two numbers are sufficient to describe the shape of a triangle. That the shape sphere is actually a sphere is harder to understand and requires that we allow triangles to degenerate, which is to say, we allow “triangles” consisting of three vertices that all lie on the same line to be called triangles. These so-called degenerate zero-area triangles form the equator of the shape sphere: they are the eclipses!
The area of a triangle, divided by its size (r) squared, is its distance to the equator. The north and south poles of the sphere represent those triangles of maximum possible area and are the two equilateral triangle shapes. But why are there two equilateral shapes? These two equilateral triangle shapes differ by the cyclic order of their vertices (see Figure 9). There is no way to turn one of these equilateral triangles into the other by a rotation, translation or scaling of the plane: they represent different shapes. Yet the operation of reflecting about a line (any line) in the plane will turn one equilateral triangle shape into the other one. This reflection operation acts on all triangles, and so on the shape sphere itself, where it acts by reflection about the equator, keeping the points of the equator (degenerate triangles) fixed while interchanging the north and south hemispheres.
Included among the degenerate triangles are the binary collisions: those “triangles” for which two of the three vertices lie on top of each other. There are exactly three of these binary collision triangles, labeled “12,” “23” and “13,” according to which two vertices lie on top of each other.
I can now explain how the shape sphere shows us that the three-body configuration space is topologically the same as the usual x-y-plane minus two points. We have to know that the sphere minus a single point is topologically the same object as the usual x-y-plane (see Figure 10). One way to see this fact about the sphere is to use stereographic projection, which maps the sphere with a single point removed (the “light source”) onto the usual x-y-plane. As a point on the sphere tends toward the light source, its image point on the x-y-plane moves out to infinity, so we can also say that the plane with a point at infinity added is topologically equal to the sphere. Take the light source to be the 13 binary collision point of the shape sphere, so that the point at infinity of the x-y-plane corresponds to the 13 collision point. Orient the sphere so that its equatorial plane intersects the x-axis of the x-y-plane. Then stereographic projection maps the equator of degenerate triangles to the x-axis of the plane and the other two binary collision points get mapped to two points on this x-axis. In this way, we arrive at exactly the picture described earlier.
The three binary collision points form three special points on the shape sphere. Besides these three there are additional special points on the shape sphere called central configurations. These five central configurations correspond to the five families of solutions discovered by Euler and Lagrange. Their solutions are the only three-body solutions for which the shape of the triangle does not change as the triangle evolves! In the Lagrange solutions, the triangle remains equilateral at each instant; there are two Lagrange configurations, as we have seen, and they form the north and south poles of the shape sphere (see Figure 11). We label them “Lagrange point 4” and “Lagrange point 5.” The remaining three central configurations are the Euler configurations, labeled “Euler point 1,” “Euler point 2” and “Euler point 3.” They are collinear (all in a line), degenerate configurations, so they lie on the equator of the shape sphere. They are positioned on the equator between the three binary collision points (see Figure 12). (Their spacing along the equator depends on the mass ratios between the three masses of the bodies.) Euler point 1, for example, lies on the equatorial arc marked 1, so is a collinear shape in which body 1 lies between bodies 2 and 3. (Often all five central configuration points are called Lagrange points, with the Euler points labeled “L1,” “L2” and “L3.”)
One can understand the central configuration solutions by dropping three bodies, by which I mean, by letting the three bodies go from rest, with no initial velocity. Typically when one does this, all kinds of crazy things will happen: close binary collisions, wild dances and perhaps the escape of one body to infinity. But if one drops the three bodies when they are arranged in one of the five central configuration shapes, then the triangle they form simply shrinks to a point, remaining in precisely the same shape as it started, with the three masses uniformly pulling on one another until the solution ends in a simultaneous triple collision.
The five roads to triple collision
Triple collision is an essential singularity within the three-body problem, something like a big bang at the center of the problem, and it is the source of much of its chaos and difficulty. In the early 1900s Finnish mathematician Karl Sundman proved that the five central configurations, as represented by the dropped solutions just described, are the only roads to triple collision. What this means is that any solution that ends in triple collision must approach it in a manner very close to one of these five dropped central configuration solutions, and as it gets closer and closer to triple collision, the shape of the solution must approach one of the five central configuration shapes.
Sundman’s work was a complicated feat of algebra and analysis. Then, in the year I graduated from high school (completely oblivious to the three-body problem), American mathematician Richard McGehee invented his so-called blow-up method, which allowed us to understand Sundman’s work pictorially and to study dynamics near triple collision in much greater detail. Let r denote the distance to triple collision—a measure of the overall size of a triangle. As r approaches zero, Newton’s equations become very badly behaved, with many terms going to infinity. McGehee found a change of configuration space variables and of time that slows down the rate of approach to triple collision and turns the triple collision point, which is r = 0, into an entire collection of points: the collision manifold. Surprise! The collision manifold is essentially the shape sphere. McGehee’s method extended Newton’s equations, originally only valid for r greater than zero, to a system of differential equations that makes sense when r = 0.
Newton’s equations have no equilibrium points, meaning there are no configurations of the three bodies that stand still: three stars, all attracting one another, cannot just sit there in space without moving. But when Newton’s equations are extended to the collision manifold, equilibrium points appear. There are exactly 10 of them, a pair for each of the five central configuration points on the shape sphere. One element of a pair represents the end result of the corresponding dropped central configuration in its approach to triple collision. Newton’s equations stay the same even if we run time backward, so we can run any solution in reverse and get another solution. When we run a dropped central configuration solution backward, we get a solution that explodes out of triple collision, reaching its maximum size at the dropped configuration. The other element of the pair represents the initial starting point of this “exploding” solution. Together these two central configuration solutions—collision and ejection—fit smoothly and form a single ejection-collision solution that leaves the ejection equilibrium point at r = 0, enters into the r greater than zero region where it achieves a maximum size, and then shrinks back to end up on the triple collision manifold at the collision equilibrium point there. This complete solution connects one element of an equilibrium pair to the other.
By creating these equilibrium points associated with central configurations, buried deep inside the three-body problem, McGehee gave Moeckel a key that enabled him to apply recently established results from modern dynamical systems—results unavailable to Newton, Lagrange or Sundman—to make some interesting headway on the three-body problem.
Moeckel’s walk
In Moeckel’s papers I saw a picture of a graph with five vertices labeled by the central configurations and joined together by edges (see Figure 13).
A walk on a graph is a possible circuit through its vertices, traveling the edges from vertex to vertex. Moeckel proved that any possible walk you can take on his graph corresponds to a solution to the three-body problem that comes close for some time to the central configuration solution labeled by the corresponding vertex. For example, the walk E1 L4 E2 L5 corresponds to a solution very close to the Euler ejection-collision solution associated with the Euler point 1, then comes close to triple collision almost along the Lagrange L4 central configuration solution, but before total triple collision is achieved the three bodies shoot out along one of the five “roads” very near to the Euler point 2 central configuration solution. Then, finally, as this Euler solution collapses back toward triple collision, the solution spins out into a Lagrange L5 equilateral shape. Moreover, if we repeat this same walk, making it periodic, the solution following it will be periodic.
Soon after Simó told me there had to be a dynamical mechanism, I realized that Moeckel’s graph embedded into the shape sphere (see Figure 14). The important thing about this embedded graph is that it carries all of the topology of the sphere with its three binary collision holes. Indeed, we can deform the thrice-punctured sphere onto the graph and in so doing turn any loop in the punctured sphere to a walk on the graph. To see this deformation, imagine the sphere as the surface of a balloon. Make three pin pricks in it, one at each binary collision hole. The balloon is made of very flexible material, so we can stretch out our three pinpricks, enlarging them until the edges of the three holes almost touch each other and the remaining material forms a ribbon hugging close to the embedded graph. In the process of making this deformation, any closed loop in the thrice-punctured sphere gets deformed into a closed loop in this ribbon structure and, from there, to a walk on Moeckel’s embedded graph.
To turn this picture into a theorem about solutions, I needed to prove that if I project the solutions guaranteed by Moeckel’s theorem onto the shape sphere, then they never stray far from this embedded graph. If they did, they could wind around the binary collisions or even hit one, killing or adding some topologically significant loops and so changing the eclipse sequence. I e-mailed Moeckel to ask for help. He wrote back, “You mean you’re going to force me to read papers I wrote over 20 years ago?” Nevertheless, he dove back into his old research and proved that the projections of the solutions he had encoded symbolically all those years ago never did stray far from the embedded graph. My question was answered—almost.
To make his proof work, Moeckel needed a tiny bit of angular momentum. (Angular momentum, in this context, is a measure of the total amount of “spin” of a system and is constant for each solution.) But for those 17 years before my conversation with Simó I had insisted on solutions having zero angular momentum. This insistence arose because solutions that minimize action among all curves having a given eclipse sequence must have zero angular momentum. On the other hand, Moeckel needed a small bit of angular momentum to get solutions traveling along the edges of his graph. The symbol for a tiny positive quantity in mathematical analysis is an epsilon. We needed an epsilon of angular momentum.
There was another catch to Moeckel’s results: his solutions, when they cross the equator of the shape sphere near the Euler points E1, E2 and E3, will oscillate back and forth there across the equator before traveling up to the north or south pole as they go in near triple collision along the corresponding Lagrange road, L4 or L5. To account for these oscillations, take a positive integer N and call an eclipse sequence “N-long” if every time a number occurs in the sequence it occurs at least N times in a row. For example, the sequence 1112222333332222 is 3-long, but it is not 4-long, because there are only three 1s in a row.
Here, finally, is our main theorem: Consider the three-body problem with small nonzero angular momentum epsilon and masses within a large open range. Then there is a large positive integer N with the following significance. If we choose any eclipse sequence whatsoever—which is N-long—then there is a corresponding solution to our three-body problem having precisely this eclipse sequence. If that sequence is made to be periodic, then so is the solution realizing it.
What about my original question? There was no large N mentioned there. I had asked about every eclipse sequence. But I did not tell you my real question. What I really wanted to know was whether or not I could realize any “topological type” of periodic curve, not any eclipse sequence. I was using the eclipse sequence as a convenient shorthand or way of encoding topological type, which is to say as a way of encoding the winding pattern of the loop around the three binary collision holes. The eclipse sequence representation of the topological type of a closed curve has redundancies: many different eclipse sequences encode the same topological type of curve. Consider, for example, the topological type “go once around the hole made by excluding the binary collision 23.” The eclipse sequence 23 represents this topological type. But so do the eclipse sequences 2223, 222223 and 2333. Whenever we have two consecutive crossings of the arc 2, we can cancel them by straightening out the meanders, making the curve during that part of it stay in one hemisphere or the other without crossing the equator (see Figure 15). Indeed, we can cancel any consecutive pair of the same number that occurs in an eclipse sequence without changing the topological type of closed curve represented by the sequence.
To use our main theorem to answer my real question, note that by deleting consecutive pairs I can ensure that the eclipse sequence that encodes a given topological type never has two consecutive numbers of the same type in it: no “11” or “22” or “33.” Call such a sequence an admissible sequence. Now, take any admissible sequence, for example, 123232. Allow me to use exponential notation in writing down eclipse sequences, so, for example, 13 = 111. Choose an odd integer n at least as big as the number N of our main theorem. Replace the admissible sequence by the longer sequence 1n 2n 3n 2n 3n 2n and continue it periodically. This longer sequence represents the same originally chosen topological type because n is odd. Our theorem says that this longer sequence is realized by a periodic solution. This periodic solution represents our original topological type 123232.
What’s Next?
We still have much left to do. When I originally posed my question almost 20 years ago, I only wanted solutions having zero angular momentum. But evidence is mounting that the answer to my question in the case of zero angular momentum is “no.” We have some evidence that even the simplest nonempty periodic sequence 23 is never realized by a periodic solution to the equal-mass, zero angular momentum three-body problem.
Our main question as posed here, even for angular momentum epsilon, remains open because our theorem allowed us to realize only sequences that are N-long for some large N. We have no clue, for example, how to realize admissible sequences, that is, sequences with no consecutive numbers of the same type.
At the end of the day, we may be no closer to “solving” the three-body problem in the traditional sense, but we have learned quite a lot. And we will keep at it—this problem will continue bearing fruit for those of us who are drawn to it. It turns out that new insights are still possible from one of the classic quandaries in mathematical history.
