Central configurations

• Richard Moeckel, School of Mathematics, University of Minnesota, Minneapolis, MN

A central configuration is a special arrangement of point masses interacting by Newton's law of gravitation with the following property: the gravitational acceleration vector produced on each mass by all the others should point toward the center of mass and be proportional to the distance to the center of mass. Central configurations (or CC's) play an important role in the study of the Newtonian N-body problem. For example, they lead to the only explicit solutions of the equations of motion, they govern the behavior of solutions near collisions, and they influence the topology of the integral manifolds.

Figure 1: A central configuration of five equal masses with gravitational acceleration vectors.

Equations for Central Configurations

Let \(m_i>0, i=1\dots N\) be the masses and \(q_i\in \mathbb{R}^d\) the position vectors of N point particles in d-dimensional space. The configuration of the N particles is represented by \(q = (q_1,\ldots,q_N)\in R^{Nd}\ .\)

The Basic CC Equations

Newton's law of motion for the gravitational N-body problem is

\[\tag{1} m_i \ddot q_i = F_i = \sum_{j\ne i}\frac{m_im_j(q_j-q_i)}{r^3_{ij}}\]

where \(r_{ij} = |q_i-q_j|\) is the Euclidean distance between particles i and j (the gravitational constant is normalized to G= 1). The force vector \(F_i\in R^d\) on the right-hand side can also be written as a partial gradient vector \(F_i = \nabla_i U\) where

\[\tag{2} U = \sum_{j\ne i}\frac{m_im_j}{r_{ij}}\]

is the Newtonian potential function and \( \nabla_i\) denotes the vector of partial derivatives with respect to the d components of \(q_i\ .\)

The acceleration of the i-th body is \(F_i\,/m_i\) so the condition for \(q\) to be a central configuration is

\[\tag{3} \nabla_i U = -\lambda m_i (q_i-c)\]

where

\[\tag{4} c=(m_1q_1+\ldots+m_Nq_N)/(m_1+\ldots+m_N)\in R^d\]

is the center of mass and \(\lambda\in R\) is a constant. By definition, \(q\in R^{Nd}\) is a central configuration for the masses \(m_i\) if and only if (3) and (4) hold for some constant \(\lambda\ .\)

It turns out, however, that the values of \(\lambda\) and \(c\) are uniquely determined by (3). To see this, note that translation invariance and degree -1 homogeneity of the Newtonian potential give

\[ \sum_i \nabla_i U = 0\qquad \sum_i q_i^T \nabla_i U = -U. \]

Together these give \(\sum_i (q_i-c)^T \nabla_i U = -U\) and then (3) shows that

\[\lambda = U/I \]

where

\[\tag{5} I = \sum m_i|q_i-c|^2\]

is the moment of inertia with respect to c. Thus \(\lambda >0 \) is uniquely determined. Finally, summing (3) shows that c must be the center of mass. Thus \(q\) is a central configuration for the given masses if and only if (3) holds for some \(\lambda\in R, c\in R^d\ .\)

Equivalent Central Configurations and Normalized Equations

The central configuration equation (3) is invariant under the Euclidean similarities of \(R^d\) -- translations, rotations, reflections and dilations. Call two configurations \(q, q' \in R^{Nd}\) equivalent if there are constants \(k\in R, b\in R^d\) and an dxd orthogonal matrix \(Q\) such that \(q'_i = k Q q_i + b, i=1,\ldots,N\ .\) If \(q\) satisfies (3) with constants \(\lambda, c\) then \(q'\) satisfies (3) with constants \(\lambda'= k^3\lambda, c' = c+b\ .\) So one can speak of an equivalence class of central configurations.

Translation invariance can be used to eliminate the center of mass. For configurations with \(c=0\) the central configuration equations become

\[\tag{6} -\lambda \, q_i = \sum_{j\ne i}\frac{m_j(q_j-q_i)}{r^3_{ij}}\]

and any configuration satisfying this equation has \(c=0\). Alternatively, substituting (4) into (3) leads, after some simplification, to

\[\tag{7} \sum_{j\ne i}m_jS_{ij}(q_j-q_i) = 0\qquad S_{ij} = \frac{1}{r_{ij}^3}-\lambda'\]

where \(\lambda' = \lambda/(m_1+\ldots+m_N)\ .\)

Dilation invariance can be used to normalize the size of a central configuration. The moment of inertia (5) is a natural measure of the size and setting \(I = 1\) is a popular normalization. Alternatively, one can normalize the size by fixing the value of \(\lambda\) in (6) or \(\lambda'\) in (7).

CC's as Critical Points

Central configurations can be viewed as constrained critical points of the Newtonian potential. To see this note that the basic equation (3) can be written

\[\nabla_i U + \frac{\lambda}{2}\; \nabla_i I = 0\]

where I is the moment of inertia. Thus a central configuration is a critical point of the restriction of \(U\) subject to the constraint \(I=const\ .\) From this point of view, \(\lambda/2\) is a Lagrange multiplier. If follows from (6) that central configurations with \(c=0\) are characterized as the critical point of \(U\) with the constraint \(I_0=const\) where \(I_0\) is the moment of inertia with respect to the origin (given by formula (5) with \(c=0\)).

With the normalization \(\lambda = 2\ ,\) central configurations are unconstrained critical points of \(U+I.\) Finally, with no size normalization, CC's are critical points of the dilation-invariant function \(I U^2\).

Examples

The Two-Body Problem

Any two configurations of N=2 particles in \(R^d\) are equivalent. Moreover, (7) reduces to just one equation

\[S_{12}(q_1-q_2) = 0\qquad S_{12} = \frac{1}{r_{12}^3}-\lambda'\]

which holds for \(\lambda' = r_{12}^{-3}\ .\) Thus every configuration of two bodies is central.

In this case, the possible motions are well-known -- each mass moves on a conic section according to Kepler's laws. In particular, one has a family of elliptical periodic motions ranging from circular (eccentricity \(\epsilon=0\)) to collinear (limit as \(\epsilon\rightarrow 1\,\)). The latter is an example of a total collision solution, that is, one for which all N bodies collide at the center of mass.

Symmetrical Configurations of Equal Masses

When all of the masses are equal, it is obvious from symmetry that certain configurations are central. In the plane (d=2), one can place the masses at the vertices of a regular N-gon or at the vertices of a regular (N-1)-gon with the last mass at the center (see Figure 2 ). Similarly, in space (d=3), a regular polyhedron or centered regular polyhedron are central configurations but these are possible only for special values of N. For d>3, regular polytopes are central configurations for certain values of N. Note however, that equal masses may admit other central configurations with less symmetry as in Figure 1 or with no symmetry at all as in Figure 3 .

When some but not all of the masses are equal it is again possible to look for central configurations which are symmetric under similarities of \(R^d\) which permute equal masses. For example, it is clear that the centered regular polygon is still a central configuration when the mass at the center is different from the others.

Figure 2: Seven equal masses forming a centered hexagon.

Figure 3: Asymmetric central configuration of eight equal masses.

Euler

The first nontrivial examples of central configurations were discovered by Euler in 1767, who studied the case N=3, d=1, that is, three bodies on a line (Euler (1767)). When two masses are equal, one can get a central configuration by putting an arbitrary mass at their midpoint (a centered 2-gon). For three unequal masses it is not obvious that any central configurations exist. But Euler showed that, in fact, there will be exactly one equivalence class of collinear central configurations for each possible ordering of the masses along the line.

For example, if the order is \(q_1<q_2<q_3\ ,\) one can reduce to the case \(q_1=0, q_2=1, q_3 = 1+r\) where \(r>0\ .\) Equations (7) give three equations for \(r, \lambda\) of which only two are independent. Eliminating \(\lambda\) leads to a fifth degree polynomial equation

\[\tag{8} \begin{align} (m_1+m_2)r^5 &+ (3m_1+2m_2)r^4 + (3m_1+m_2)r^3 \\ &-(m_2+3m_3)r^2-(2m_2+3m_3)r-(m_2+m_3) = 0. \end{align} \]

Descartes' rule of signs shows that there is exactly one positive real root, \(r(m_1,m_2,m_3)>0\ .\)

Figure 4: Algebraic surface defined by Euler's equation.

(8) illustrates the nature of the problem of finding central configurations. In the general case, one has a complicated set of algebraic equations for the configuration, \(q\ ,\) depending on the masses as parameters. The goal is to find or at least count the equivalence classes of real solutions. If we normalize \(m_3=1\) (8) becomes an equation in three variables \((m_1,m_2,r)\) defining a complicated algebraic surface shown in Figure 4. The horizontal variables are the mass parameters. Fixing \((m_1,m_2)\) defines a vertical line and Euler's CC is found by intersecting this line with the surface. It is remarkable that for \(m_i>0\), there is just one intersection.

Euler's result generalizes nicely to the N-body problem. For each ordering of the bodies along the line, there is exactly one collinear central configuration, a result of F.R.Moulton (Moulton (1910)).

Lagrange

Lagrange found next example in the planar three-body problem \(N=3, d=2\). Remarkably, an equilateral triangle is a central configuration, not only for equal masses, but for any three masses \( m_1, m_2, m_3 \). Moreover, it is the only noncollinear central configuration for the three-body problem (Lagrange (1772)).

When the masses are not equal, the center of mass will not be the center of the triangle and it is not at all obvious that the configuration is central. But it is easy to see it using mutual distance coordinates. The three mutual distances \(r_{12}, r_{31}, r_{23} \) can be used as local coordinates on the space of noncollinear configurations of three bodies in the plane up to symmetry. The potential and the moment of inertia can be expressed in these coordinates as

\[ U = \frac{m_1 m_2}{r_{12}}+\frac{m_3 m_1}{r_{31}}+\frac{m_2 m_3}{r_{23}}\qquad\qquad\qquad I = (m_1m_2 r_{12}^2+m_3m_1 r_{31}^2+m_2m_3 r_{23}^2)/(m_1+m_2+m_3).\]

Now use the characterization of CC's as critical points of \(U\) with fixed \(I\). Setting

\[ \frac{\partial U}{r_{ij}} + \frac{ \lambda}{2} \frac{\partial I}{r_{ij}} \]

gives

\[ r_{ij}^3 = \frac{m_1+m_2+m_3}{\lambda} \]

which holds for some \(\lambda\) if and only if the three distances are equal.

This result can be generalized to higher dimensions: the regular simplex is a CC of N bodies in N-1 dimensions for all choices of the masses and is the only CC spanning the full N-1 dimensions.

(1+N)-Body Central Configurations

Figure 5: Central Configuration with 1 Large and 3 Small Masses.

The limiting case of the (N+1)-body problem where N of the masses tend to zero is called the (1+N)-body problem and G.R. Hall asked about the limits of the corresponding CC's. If the N small masses are equal, the centered N-gon is always a CC and the limit could be viewed as a model of a symmetrical ring of small masses around a planet. In general it turns out that all of the small bodies converge to a circle around the large one, but lumpy rings like the one in Figure 5 can also occur (Casasayas-Llibre-Nunes (1994)).

An Existence Proof

The characterization as critical points provides an easy way to see that CC's exist, i.e., that the complicated set of algebraic equations (3) always has solutions. For any choice of \(N\) positive masses, the constraint set \(I_0 = 1\) is an ellipsoid in \( R^d\). The Newtonian potential defines a smooth function on the open subset where all of the distances \(r_{ij} \) are nonzero. Since the ellipsoid is compact and since \(U\rightarrow\infty\) as \(r_{ij}\rightarrow 0 \), it follows that \(U\) attains a minimum at some configuration \(q\). This will be a constrained critical point and hence a CC.

Self-Similar Solutions

Central configurations give rise to simple, explicit solutions of the N-body problem such that the configurations at any two times are similar, i.e., they differ only by rotation, translation and dilation. Each of the bodies moves according to Newton's laws but the special balance of forces characterizing CC's makes it possible for the overall shape of the N-body configuration to remain the same. Such a solution is sometimes called homographic, although self-similar might be a better term.

For N=2 the shape is just a line segment. As the bodies move on their Keplerian orbits, this line segment rotates, translates and changes size. While this is a rather trivial shape, the motions of the individual bodies is interesting. It turns out that such Keplerian motions are also possible for N>2 and these will be described below.

By taking the center of mass at the origin, the translational motion is eliminated and only rotations and dilations remain. The simplest case is a pure dilation.

Homothetic Solutions

A homothetic solution of the N-body problem is a solution of Newton's equations (1) such that \(q_i(t) = r(t)q_i(0)\) for all i, where \(r(t)>0\) is a dilation function describing how the size of the configuration changes with time. Each body moves along a line through the origin. From Newton's equations one finds that such a homothetic solution is possible if and only if the initial configuration \(q(0)\) satisfies (6) and the dilation function solves

\[\ddot r(t) = -\frac{\lambda}{r(t)^2}.\]

This means the configuration is a CC with \(c=0\) and the size function solves the one-dimensional Kepler problem".

Figure 6: Homothetic motion of Lagrange's equilateral triangle with masses 10,2,1.

The Kepler problem describes the motion of a point particle gravitationally attracted to a mass \(\lambda\) at the origin. For example, starting with zero initial velocity, it is clear that \(r(t)\) decreases to zero in a finite time. For N=2 the corresponding homothetic solution has the two bodies approaching collision at the origin along straight lines. Such a solution is possible for every CC. For example, if the bodies in Figure 1, Figure 2 or Figure 3 are released with zero initial velocity, they will move along the lines indicated by their acceleration vectors toward a collision at the center of mass. Figure 6 shows a homothetic solution based on Lagrange's equilateral triangle. The masses are 10, 2, 1 so the center of mass is not the center of the triangle. During the motion, the triangle remains equilateral as the bodies move along straight lines, at increasing speed, toward triple collision.

Homographic Solutions and Relative Equilibria in the Plane

Consider the N-body problem in the plane \(\mathbb{R}^2\). The most general homographic solution has

\[q_i(t) = r(t)\,Q(\theta(t)\,)q_i(0)\qquad Q(\theta) = \begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\]

for some dilation \(r(t)\) and some rotation angle \(\theta(t)\). Substituting into Newton's laws one finds that the initial configuration \(q(0)\) is a CC and that \(r(t),\theta(t)\) must solve the two-dimensional Kepler problem.

\[\ddot r(t) -r(t)\dot\theta(t)^2= -\frac{\lambda}{r(t)^2}\qquad r(t)^2\dot\theta(t) = const.\]

The solutions give the familiar Keplerian conic sections when N=2 and analogous solutions are possible for any planar CC.

In particular, every planar CC gives rise to a family of periodic solutions of the N-body problem such that each of the bodies moves on a Keplerian elliptical orbit. As a special case, the circular Keplerian orbits give rise to a homographic solution for which the configuration just rotates rigidly at constant angular speed. This is called a relative equilibrium solution and planar CC's are often called relative equilbria (RE's). These are true equilibrium solutions in a uniformly rotating coordinate system or in a rotation-reduced system where the rotational symmetry of the N-body problem has been eliminated. Figure 7 shows a relative equilibrium solution based on one of Euler's collinear CC's for the three-body problem. The masses are 81, 1, 0.1 corresponding roughly to the Earth, Moon, and some smaller body. The small mass is permanently eclipsed by the Moon and would be invisible from the Earth.

The relative equilibrium solutions correspond to elliptical orbits of the Kepler problem of eccentricity \(e=0\). Eccentricities \(0<e<1\) give rise to solutions where \(r(t)\) and the angular speed \(\dot\theta(t)\) are not constant. The limiting case \(e=1\) gives the planar homothetic solutions described above. Figure 8 shows a homographic solution with \(e=0.8\) based on the CC from Figure 3. The strange shape is preserved while the size and rotation angle change in Keplerian fashion.

Figure 7: Eulerian Relative Equilibrium

Figure 8: Homographic Motion with Eccentricity 0.8

Higher-Dimensional Homographic Motions

For the N-body problem in the physically relevant dimensions \(d\le 3\) the homographic motions described above exhaust all the possibilities. In other words, any CC can exhibit a homothetic collapse and any planar CC can perform planar Keplerian homographic motions and that is all. In particular, a nonplanar CC such as the regular tetrahedron can only manage a homothetic motion and a non-CC cannot participate in any homographic motions at all. However, recent work has shown that new possibilities arise if we allow dimensions \(d\ge 4\) (Albouy-Chenciner (1998), Chenciner (2011), Chenciner-Jiménez-Pérez (2013)).

The new phenomena are due to the greater complexity of rotations in higher dimensions. For example, in \(\mathbb{R}^4\) it is possible to rotate in two mutually orthogonal planes with different angular speeds. This leads to new ways of balancing the gravitational forces with centrifugal forces to get rigid motions (relative equilibrium solutions). There is a wider class of balanced configurations (BC's) which strictly includes the CC's giving rise to such four-dimensional rigid motions. For example, a nonequilateral and non collinear isosceles triangle is not a CC but is a BC if the two symmetric masses are equal. Figure 9 show a three-dimensional projection of a four-dimensional rigid motion of such a triangle. If the ratio of the rotation speeds in the two orthogonal planes is irrational, the motion is quasiperiodic.

To get a nonrigid homographic solution it is still necessary to have a CC and the motion will still feature planar Keplerian ellipses. However, the ellipse corresponding to different bodies may lie in different planes. Some impression of this kind of motion can be gotten from Figure 10 which shows a solution based on the regular tetrahedron with four equal masses. The solution is a rigid motion in \(\mathbb{R}^4\) so the configuration is always a regular tetrahedron. The animation shows a three-dimensional projection in which the shape appears to change, but it is clear that each body is moving on its own circle in distinct planes. Elliptical orbits of the same kind are readily imagined. Thus the tetrahedron, which can exhibit only homothetic motions in \(\mathbb{R}^3\) has a much larger repertoire of homographic motions in \(\mathbb{R}^4\).

Figure 9: Rigid Motion of an Isosceles Triangle (BC) \(\mathbb{R}^4\).

Figure 10: Rigid Motion of a Tetrahedron (CC) in \(\mathbb{R}^4\).

Stability of Relative Equilibria

Every planar CC determines a relative equilibrium solution of the N-body problem with every body moving on a circle with constant angular speed. In a uniformly rotating coordinate system these become equilibrium solutions and one can analyze their linear stability by finding the eigenvalues of the linearized differential equations. Strictly speaking, even the circular periodic orbits of the two-body problem are linearly unstable because they are part of a four-parameter family of RE solutions with different sizes, centers of mass and rotation speeds. This applies to all relative equilibria and is reflected in the existence of four eigenvalues with value \(0\) organized into two Jordan blocks. This trivial drifting apart of nearby relative equilibria can be eliminated by fixing the center of mass, energy and angular momentum. Then for linear stability it is necessary that the rest of the eigenvalues be purely imaginary numbers.

With this understanding, it is known (Gascheau (1842)) that Lagrange's equilateral triangle solutions are linearly stable provided the masses satisfy

\[27(m_1m_2+m_3m_1+m_2m_3)<(m_1+m_2+m_3)^2.\]

This holds only if there is a dominant mass, i.e., one mass must be much larger than the other two. It does hold in the case of the Sun, Jupiter and a third small body so, ignoring the other planets, a small body forming an equilateral triangle with the Sun and Jupiter could remain there for a long time. In fact there are clusters of small bodies found near these equilateral points: the Greek and Trojan asteroids. On the other hand, the collinear Eulerian relative equilibria are always linearly unstable. For example, the solution of Figure 7 is unstable and nearby solutions will diverge from it exponentially.

The RE based on the regular N-gon with N equal masses is linearly unstable as is the centered N-gon with N+1 equal masses as in Figure 2. On the other hand a regular N-gon with a sufficiently large (N+1)-st mass at the center is linearly stable, provided N is at least 7 (Moeckel (1995)). For small N, lumpy rings like the one in Figure 5 can be linearly stable.

James Clerk Maxwell studied the centered N-gon in connection with the stability of the rings of Saturn. He found that the mass ratio guaranteeing stability increases with N, so a given ring mass should not be split into too many pieces. This supported the idea that the rings were composed of discrete bodies instead of a continuously distributed dust or liquid. (Maxwell (1859)).

Other Applications of Central Configurations

Central configurations play several other roles in the N-body problem besides providing self-similar solutions.

Collisions

Central configurations are important for the study of collisions in N-body problem. The homothetic motions of central configurations as in Figure 6 provide examples of total-collision orbits -- at a certain moment, all of the bodies collide at the center of mass. Although there exist other nonhomothetic total-collision orbits, all such orbits approach central configurations at collision. More precisely, if the collapsing configuration is blown-up, say to have moment of inertia 1, then the rescaled configuration approaches the set of central configurations. For example, in the three-body problem, Siegel showed that every triple collision solution has asymptotic shape either an equilateral triangle or one of Euler's collinear CC's (Siegel-Moser (1971), McGehee (1978)).

For more general types of collisions, Von Zeipel's theorem states that there is a well-defined limiting position for each body (McGehee (1986)). The bodies with the same limiting position are said to form a cluster. As the solution approaches collision the shape of each cluster approaches the set of central configurations (Saari (1984)).

Bifurcations of Integral Manifolds

The phase space of the N-body problem in \(\mathbb{R}^d\) is the space of all possible position and velocity vectors of the bodies, so it has dimension \(2dN\). Each solution determines a curve \(p(t)\) in phase space. An integral manifold is a subset of phase space obtained by fixing the values of the well-known constants of motion (or integrals): the energy, angular momentum, total linear momentum and center of mass. If the gradients of these functions are linearly independent at a certain point of phase space then fixing their values defines a submanifold. For the physically relevant cases \(d=2,3\) these are submanifolds of dimension \(4N-6, 6N - 10\) respectively.

The points where the gradients are not linearly independent are the singular points of the energy-momentum mapping and the corresponding values of the integrals are the singular values. Fixing the constants at such a singular value generally gives a subset of phase space which is not a manifold near the singular points. Furthermore, different choices of the constants near the singular value can lead to nonsingular integral manifolds with different topological structures. In other words the singular points and singular values are associated with bifurcations of the topology of the integral manifolds.

In the physically relevant cases, the singular points are just the phase space points associated to relative equilibrium solutions. In other words, the position vectors at a singular point form a planar central configuration and the velocities are those of the corresponding rigid motion solution. Due to the noncompactness of the phase space, other kinds of bifurcations at infinity also have to be considered to get a complete description of the possible topologies. (Smale (1970,1), Cabral (1973), Albouy (1993)).

Perturbations

The homographic motions associated to a CC are very special solutions of the N-body problem but perturbation theory can be used to find many other solutions nearby. Near the relative equilibria there are other periodic solutions or quasi-periodic solutions. Examples include the Trojan asteroid orbits and the motion of perturbed polygonal rings (Meyer-Schmidt (1993), Chenciner-Fejoz (2009)).

An unstable relative equilibrium has stable and unstable manifolds consisting of solutions which converge to the relative equilibrium solution as time tends to \(\pm\infty\). In the case of the Eulerian, collinear CC's of the three-body problem, these manifolds act as separatrices governing the motion of orbits from one part of the phase space to another and can even be relevant for practical space mission design (Conley (1968), Belbruno-Miller (1993), Koon-Lo-Marsden-Ross, Gomez-Llibre-Martinez-Simó ). In this way the central configurations can have more influence over the global dynamics than might be expected at first glance.

Counting Central Configurations

Central configurations are defined by a complicated system of algebraic equations (3) involving both the positions and the masses of the N bodies. The solution set of these equations is an algebraic subset of the product space of the mass space and the position space. Even for the simple case of the collinear three-body problem, this algebraic set is quite complicated as shown in Figure 4. For fixed values of the masses, one would like to find all of the corresponding CC's. In Figure 4 this would amount to finding the points of intersection of the surface with each fixed vertical line. The fact that there is no tractable solution to Euler's fifth-degree equation (8) makes it clear that solving the general CC equations is hopeless. A more reasonable goal for theoretical studies of central configurations is to prove something about the total number of CC's and to attempt some kind of classification of the solutions. In addition to its inherent interest, the CC problem is a kind of paradigm for problems involving complicated systems of real, algebraic equations.

Finiteness Problem

A basic question to ask about any system of algebraic equations is just whether the number of solutions is finite. If it's not finite one could imagine an infinite set of point solutions, a curve of solutions or a higher-dimensional set of solutions. For the CC equations, this is sometimes called the Chazy-Wintner-Smale problem: given N positive masses, is the number of central configurations finite, up to symmetry ? Smale singled out the planar case as the \(6\)-th of his problems for the twenty-first century. The phrase up to symmetry is important since the set of CC's is invariant under rotations, translations and dilations.

The characterization of CC's as constrained critical points of the Newtonian potential suggests that the answer should be Yes. A typical smooth function on a compact manifold (like the ellipsoid of fixed moment of inertia) has a finite number of critical points. Although the Newtonian potential is only smooth on the noncompact open set of noncollision configurations, it is known that CC's avoid a neighborhood of the collision set (Shub (1970)). Working on the ellipsoid, there is still a rotational symmetry, but it is also true that a generic smooth, rotation-invariant function has finitely many rotational equivalence classes of critical points. So the answer to the Chazy-Wintner-Smale question is expected to be Yes, at least for generic choices of the masses. However, actually proving finiteness or even generic finiteness is difficult.

For N=3 there are exactly five CC's for every choice of positive masses \(m_1,m_2,m_3\), counted up to translations, rotations and dilations in the plane. These are the three Eulerian collinear CC's, one for each rotationally distinct ordering of the three bodies along the line and two Lagrangian equilateral triangles, distinguished by whether the three masses occur in clockwise or counterclockwise order around the triangle. If rotations in \(\mathbb{R}^3\) are allowed these two equilateral solutions become equivalent and the total number is four.

For N=4, the number of CC's depends on the choice of masses. The regular tetrahedron is the only nonplanar CC and according to Moulton's theorem, there are exactly 12 collinear CC's. It is known that for any four positive masses, the number of symmetry classes of planar CC's is finite and lies between 32 and 8472 where the count is up to rotation, translation and dilation in the plane (Hampton-Moeckel (2006)). The lower bound is known to occur while the true upper bound is probably 50, which occurs for the case of four equal masses (Simo (1978), Albouy (1996)). It is also known that there are at least six planar CC's in the shape of a convex quadrilateral, at least one for each cyclic ordering if the bodies (Xia (2004)).

For N=5, the number of CC's is known to be finite for generic choices of the five masses (Albouy-Kaloshin (2012), Moeckel (2001)). The regular simplex is a CC in \(\mathbb{R}^4\) and there are 60 collinear Moulton CC's. There are some fixed but unrealistic bounds on the number of planar and three-dimensional CC's. It is also known that the positivity condition on the masses is necessary for finiteness. In fact, for the nonpositive mass vector \((-1,4,4,4,4)\) there is a curve of solutions (Roberts (1999)).

For N>5, even generic finiteness is an open problem.

Morse Theory

Morse theory relates the topology of a smooth manifold to the critical points of smooth functions on the manifold. For generic smooth functions, each critical point is nondegenerate and can be assigned a nonnegative integer Morse index describing the local behavior of the function. For example, on a two-dimensional manifold, critical points are classified as minima (index \(0\)), maxima (index \(2\)) or saddle points (index \(1\)). The critical point data about a function can be stored in a generating function

\[M(t) = \gamma_0 +\gamma_1\,t + \gamma_2\,t^2+\ldots\]

where \(\gamma_k\) is the number of critical points of index \(k\). The topology of the manifold is represented by its Betti numbers \(\beta_k, k = 0,\ldots, dim \) describing (roughly) the number independent of k-dimensional cycles. The Poincaré polynomial is the generating function

\[P(t) = \beta_0 +\beta_1\,t + \beta_2\,t^2+\ldots.\]

For a generic smooth function, the Morse inequalities are summarized by the equation

\[\tag{9} M(t) = P(t) + (1+t)R(t)\]

where \(R(t) = r_0+ r_1\, t+\ldots\) is some polynomial with nonnegative coefficients.

Morse theory for the planar N-body problem was initiated by Smale and studied by Palmore (Smale(1970),Palmore(1973-75)). The manifold is obtained by deleting the collision set from the ellipsoid of constant moment of inertia and then taking a quotient space under the rotational symmetry. The Poincaré polynomial turn out to be

\[P(t) = (1+2t)(1+3t)\ldots (1+(N-1)t).\]

For N=3 the quotient manifold is a two-sphere with three deleted points and Poincaré polynomial \(P(t) = 1+2t\). There are 3 saddle points (the Euler CC's) and 2 minima (the Lagrange CC's) and the Morse inequalities are satisfied with \(R(t)=1\)

\[2+3t = (1+2t) + (1+t)1.\]

For N=4, the Poincaré polynomial is \(P(t) = (1+2t)(1+3t) = 1+5t + 6t^2\). Assuming that the critical points are nondegenerate, there are at least 6 minima (the convex CC's) and at least 12 saddles of index 2 (the collinear CC's). It is also known that all critical points in this case have index at most 2. Using \(\gamma_0\ge 6, \gamma_2\ge 12\) in (9) shows that \(r_0\ge 5, r_1\ge 6\). Putting this back into (9) gives

\[\gamma_0 +\gamma_1\,t + \gamma_2\,t^2 \ge 1+5t+6t^2 + (1+t)(5+6 t).\]

Setting t=1 gives an estimate for the total number of critical points \(\gamma_0 +\gamma_1 + \gamma_2\ge 34.\) This seems to be a sharp lower bound for the nondegenerate case, the true lower bound of 32 being achieved at a bifurcation point.

For general N, the Morse estimate for the number of symmetry classes of planar CC's is \(\frac{(3N-4)N!}{2}\) including the \(\frac{N!}{2}\) collinear ones.

Morse theory for the three-dimensional problem is more complicated since it is not possible to simply quotient out the rotation group.

Many Small Masses

Instead of the (1+N)-body problem as described above, Xia considered the case of 2 or 3 large masses and many small ones (Xia (1991)). By adding one small mass, then another even smaller mass and so forth, he was able to inductively construct all of the central configurations for certain open sets of positive masses.

Starting from two large masses in the plane there are exactly five places to put another, negligibly-small mass to get a CC. Namely, the third mass must go at one of the two equilateral triangle positions or one of the three limiting cases of Euler's collinear configurations. These five critical positions are found as critical points of a certain potential function based on the two pre-existing large masses. Using the implicit function theorem, it can be shown that each of these five critical positions gives rise to a family of CC's of the three-body problem defined for sufficiently small positive values of the third mass.

Now having chosen where to put the third small mass, one can look for possible positions of a fourth even smaller mass. These are the critical points of a new potential function determined by the first three masses. If the third mass was placed at an equilateral position, there will be four new critical points of this potential function nearby. Together with the four remaining critical points from the first step, there are eight places to put the fourth mass. On the other hand if the third mass was placed at a collinear critical position, there are only two new critical positions created nearby, for a total of six places to put the fourth mass. Continuing in this way one can find a recurrence formula for the total number of CC's for masses of this type. Solving it one finds that the number of equivalence classes of CC's is exactly \[(N-2)!\left((N-2)2^{N-1} + 1\right) \] for a certain open set of 2 large and many small masses. It is interesting that these numbers are much larger than the Morse-theory estimates. Starting from 3 large masses and proceeding in a similar way gives a different count. It follows that for all N>3, the number of CC's depends on the choice of masses.

One can also take an inductive approach to the spatial N-body problem with the result that for a certain open set with two large and many small masses the total number of equivalence classes of spatial CC's is exactly. \[\frac{(N-2)!}{16} \left((2N+3)3^{N-1} + 2N^2-6N-1\right) \] for a certain open set of 2 large and many small masses. For these masses the number of spatial CC's is much larger than the number of planar ones.

Recommended Reading

More information about central configurations can be found in the expository references: Moeckel (1994,2014), Saari (1980, 2005, 21st).

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Sponsored by: Prof. Alessandra Celletti, Dipartimento di Matematica, Universita' di Roma (Tor Vergata), Italy

Reviewed by: Dr. Alain Chenciner, Math Dept Paris 7 University and IMCCE (Paris Observatory), France

Reviewed by: Dr. Winston Sweatman, Institute of Natural & Mathematical Sciences - Massey University, Albany, New Zealand

Accepted on: 2014-04-01 23:05:52 GMT