Continuations and bifurcations of relative equilibria for the positive curved three body problem

A relative equilibrium for the Newtonian $n$–body problem, is a particular solution where the masses are rotating uniformly around their center of mass with the same angular velocity. In these kind of motions the masses preserve their mutual distances for all time, that is, they behave as a rigid body motion. The fixed configuration at any time is called a central configuration. In the corresponding rotating frame they form an equilibrium point, from here the name [22, 17].

Since the central configurations are invariant under rotations and homotheties, we count classes of central configurations modulo these Euclidean transformations; then for $n=3$, there are only 5 classes of relative equilibria, three collinear or Euler relative equilibria [10] and two equilateral triangle or Lagrange relative equilibrium [16].

The curved $n$–body problem is a natural extension of the classical Newtonian problem to spaces of constant curvature $\kappa$ which could be positive or negative. For $n=2$, the problem is divided into two classes, the Kepler problem (one particle is fixed, and the other one is moving according to it) and the $2$–body problem (both masses are moving according to their mutual attractions). The first one is an old problem, introduced independently in the 1830’s by J. Bolyai and N. Lovachevsky the codiscovers of the first non-Euclidean geometries. The second one was introduced by Borisov et al [4, 5]. Unlike the Newtonian problem, on the sphere these problems are not equivalent, the first one is integrable and the second one is not [20].

In 1994, V.V. Kozlov showed that the cotangent potential is the natural way to extend the Newtonian potential to spaces of constant positive curvature [13]. In 2012, F. Diacu, E. Perez-Chavela and M. Santoprete [6], obtained the generalization of this problem in an unified way for any value of $n$ and any value of the constant curvature $\kappa$. You can see [7] and [5] for a nice historical description of this problem.

In this paper we are interested in the analysis of relative equilibria for the two dimensional positive curved three body problem, which can be reduced to the analysis on the unit sphere $\mathbb{S}^{2}\subset\mathbb{R}^{3}$. That is, we will study relative equilibria for three positive masses moving on $\mathbb{S}^{2}$ under the influence of the cotangent potential. From here on, just to simply the notation we call to the relative equilibria simply as $RE$.

By exploiting the symmetries of some configurations and using spherical trigonometric arguments, several authors have found different families of $RE$ on the sphere $\mathbb{S}^{2}$ see for instance [6, 7, 8, 9, 18, 19, 21, 23].

In a recent paper [11], the authors of this article developed a new systematic geometrical method to study $RE$ on $\mathbb{S}^{2}$, where the masses are moving on the sphere under the influence of a potential which only depends on the mutual distances among the masses, in particular for the cotangent potential. For $n=3$ the authors divide the analysis of $RE$ on $\mathbb{S}^{2}$ in two big classes, the Euler relative equilibria ($ERE$ by short) where the three masses are on the same geodesic, and the Lagrange relative equilibria ($LRE$ by short), for the $RE$ which are not in the previous class. In the same paper [11], the authors find the necessary and sufficient conditions on the shapes, to obtain $ERE$ and $LRE$. In this paper we will restrict our analysis to the cotangent potential, that is, to the positive curved three body problem.

In [1], the authors proved that any $RE$ of the planar $n$–body problem can be continued to spaces of constant curvature $\kappa$, positive or negative for small values of the parameter $\kappa$. This is a remarkable result, because for instance, it is well known that any three masses located at the vertices of an equilateral triangle generate a $RE$ on the plane; however in the case of curved spaces, $LRE$ with equilateral triangle shape only exist if the three masses are equal. In particular they show that any Lagrange relative equilibria can be continued to the sphere (the equilateral triangle shape, is not preserved in the continuation if the masses are not equal).

In this paper we will show that in general, we can continue a $RE$ on the sphere, represented by an one dimensional curve. When two continuations of $RE$ intersect, we call it a bifurcation, in the next section we will give the precise definition of these concepts.

After the introduction, the paper is organized as follows: In Section 2 we give the definitions of all concepts used along the paper, and we show how use the implicit function theorem to find the bifurcation points and the extensions of solutions. In Section 3 we do the analysis for the case of equal masses and in Section 4 we do the analysis for the case of partial equal masses. In section 5, we study $LRE$ with general masses, and finally in Section 6 we summarize our results and state some final comments.

We consider three positive masses denoted by $m_{k}$ moving on a sphere of radius $R$ that we denote as $\mathbb{S}^{2}$. The equations of motions are described by the Lagrangian, which in spherical coordinates $(R,\theta_{k},\phi_{k})$ take the form,

To facilitate the notations, we set $R=1$ unless otherwise specified. We denote the angle of the minor arc on the great circle connecting the masses $i$ and $j$ as $\sigma_{ij}$. In order to avoid singularities, we exclude the case $\cos^{2}\sigma_{ij}=1$, which corresponds to $\sigma_{ij}\neq 0,\pi.$

The above angles are related to each other by the relationship

For the three-body problem, it is convenient to use the notation $\sigma_{k}=\sigma_{ij}$ for $(i,j,k)=(1,2,3),(2,3,1),$ and $(3,1,2)$. We define

The inequalities in $U_{\textrm{phys}}$ are the conditions of $\sigma_{k}$ to form a triangle. Note that one point in $U_{\textrm{phys}}$ corresponds to two triangles with opposite orientation.

As we have seen in the previous Section, there are two kinds a $RE$ on the sphere, $ERE$ and $LRE$. In [11], we proved that the $ERE$ must be on the equator or on a rotating meridian.

The big difficulty to study $RE$ on the sphere is that the linear momentum and the center of mass are not more a first integral for the positive curved problem.

Fortunately in [11], we found that the center of mass can be substitute by the vanishing of two components of the angular momentum on the sphere.

To verify this fact, we observe that the Lagrangian is invariant under rotations around the centre of $\mathbb{S}^{2}$, the angular momentum vector ${\bf c}=(c_{x},c_{y},c_{z})$ is a first integral. The components of ${\bf c}$ are

Then, fixing the rotation axis as the $z$–axis, we obtain that the components $c_{x}=0$ and $c_{y}=0$ are integrals of motion.

Now by Definition 1, after the substitution $\dot{\theta}_{k}=0$ and $\dot{\phi}_{k}(t)=\omega$, the angular momentum has the form ${\bf c}=(c_{x},c_{y},c_{z})=(0,0,c_{z}),$ where

We observe that taking $r_{k}=R\theta_{k}$ finite, in the limit $R\to\infty$ we obtain $\theta_{k}\to 0$, since $\theta_{k}=r_{k}/R\to 0$ for $R\to\infty$. Then, using equation (4), the expansion for $\cos\theta_{k}$ and $\sin\theta_{k}$ and dropping the higher order terms we obtain

Finally using the fact that $(c_{x},c_{y})=(0,0)$, the above equation (5), is the condition to fix the center of mass at the origin on the Euclidean plane (see [11] for more details).

Using the two integrals $c_{x}=0$ and $c_{y}=0$, we prove that the problem to find $RE$ on $\mathbb{S}^{2}$ is reduced to solve the eigenvalue problem $J\Psi_{L}=\lambda\Psi_{L}$ where $J$ is an useful representation of the inertia tensor given by (see [11] for more details)

In the same paper [11], we show that the necessary and sufficient conditions to have $LRE$ or $ERE$, can be described only in terms of $m_{k}$ and $\sigma_{k}$, $k=1,2,3$. The conditions for a shape to form $LRE$ are

Let $\lambda_{ij}$ be the difference of $\lambda_{i}$ and $\lambda_{j}$, namely

Then, the condition for have a $LRE$ is equivalent to $\lambda_{12}=\lambda_{23}=0$.

The correspondence between the solution $\sigma_{k}\in U_{\textrm{phys}}$ of this condition and the configuration variables $\theta_{k}$ is given by

Then using $\sigma_{k}$ and $\theta_{k}$, equation (2) determines $\cos(\phi_{i}-\phi_{j})$. Where $M=\sum_{\ell}m_{\ell}$ is the total mass, and $s=\pm 1$. If we take $s=1$ the three masses are on the northern hemisphere, when $s=-1$ they are on southern hemisphere. Finally, the angular velocity is given by

Thus, the problem to find a configuration of $LRE$ is reduced to find the solution $\sigma_{k}$ of the condition $\lambda_{12}=\lambda_{23}=0$ (See [11] for more details).

Similarly, the necessary and sufficient condition for a shape to form an $ERE$ on a rotating meridian with $\sigma_{3}=\sigma_{1}+\sigma_{2}$, namely the mass $m_{3}$ is located between $m_{1}$ and $m_{2}$, is

You can consult reference [11] for the correspondence between $\sigma_{k}$ and the configuration variables $\theta_{k}$, $\omega^{2}$.

The conditions for a shape to form an $ERE$ on the equator $\sigma_{1}+\sigma_{2}+\sigma_{3}=2\pi$ are

where $\mu_{k}=\sqrt{m_{i}m_{j}}$. For this case, $\sigma_{k}$ is given by

Note that there is just one set of $\sigma$’s (two shapes with different orientation) for given masses. For the $LRE$ and the $ERE$ on a rotating meridian on the sphere, we will show that some of these $RE$ can be continued into the three dimensional space given by $(\sigma_{1},\sigma_{2},\sigma_{3})$, and that these continuations can meet in this space. We call this “bifurcation”, to be more precise we define:

We will show ahead in this paper, that some $LRE$ meets $ERE$ on the equator, and since an $ERE$ on the equator is just one point for given mass ratio and given orientation, we define:

Finally we have that the shapes of $LRE$ can be grouping into equilateral, isosceles, and scalene triangles. We will show that for equal masses case, $m_{1}=m_{2}=m_{3}$, equilateral and isosceles $LRE$ have continuation. When only two masses are equal, for instance $m_{1}=m_{2}\neq m_{3}$, we have continuation of isosceles and scalene $LRE$. We define the bifurcation point as follows.

The existence of bifurcation points between $LRE$ one in the group $A$ and other in the group $B$ can be understood by two simple but important properties of the surfaces defined by $\lambda_{ij}=0$ in $U_{\textrm{phys}}$.

From this property we obtain the following proposition and corollaries whose proofs are obvious.

This property explains the existence of bifurcation points between two groups of shapes of $LRE$.

We explain why for $m_{1}=m_{2}\neq m_{3}$ we obtain a bifurcation point of isosceles $LRE$ and scalene $LRE$. For this case, the surface $\lambda_{12}=0$ is split into the plane $\sigma_{1}=\sigma_{2}$ , and the surface $\tilde{\lambda}_{12}=0$ by Property 2. Then, the condition for $LRE$, $\lambda_{12}=\lambda_{23}=0$, is split into two cases, intersection of $\sigma_{1}=\sigma_{2}$ and $\lambda_{23}=0$ namely isosceles $LRE$, and intersection of $\tilde{\lambda}_{12}=0$ and $\lambda_{23}=0$, namely scalene $LRE$. Then, the intersection of the three surfaces $\sigma_{1}=\sigma_{2}$ and $\tilde{\lambda}_{12}=0$ and $\lambda_{23}=0$ gives the bifurcation points.

Similarly, there are bifurcation points of equilateral $LRE$ and isosceles $LRE$ for the equal masses case, $m_{1}=m_{2}=m_{3}$.

We will use the implicit function theorem to study bifurcations. By this theorem, if two continuous differentiable functions $f(x,y,z)$ and $g(x,y,z)$ have a solution $f(x_{0},y_{0},z_{0})=g(x_{0},y_{0},z_{0})=0$, and $\nabla f\times\nabla g\neq 0$ at $(x_{0},y_{0},z_{0})$, then a continuation of the solution $f=g=0$ from this point exists in some finite interval. A geometrical interpretation of this theorem is that the conditions $f=g=0$ and $\nabla f\times\nabla g\neq 0$ means that the two surfaces intersect at this point and the vector $\nabla f\times\nabla g$ represents the tangent vector of the intersection curve at this point.

As shown in Property 2, the conditions for have a $LRE$, $\lambda_{12}=0$ and $\lambda_{23}=0$ are split into ($\sigma_{1}=\sigma_{2}$ or $\tilde{\lambda}_{12}=0$) and ($\sigma_{2}=\sigma_{3}$ or $\tilde{\lambda}_{23}=0$).

First we will show that there are no scalene $LRE$ (intersection of $\tilde{\lambda}_{12}=0$ and $\tilde{\lambda}_{23}=0$), then we describe the equilateral ($\sigma_{1}=\sigma_{2}$ and $\sigma_{2}=\sigma_{3}$) and isosceles $LRE$ ($\sigma_{1}=\sigma_{2}$ and $\tilde{\lambda}_{23}=0$ or $\sigma_{2}=\sigma_{3}$ and $\tilde{\lambda}_{12}=0$) and the bifurcation between them.

In this section, we consider the case $m_{1}=m_{2}=m_{3}\nu$, with $\nu>0$.

As shown in Property 2, the condition $\lambda_{12}=0$ is reduced to $\sigma_{1}=\sigma_{2}$ or $\tilde{\lambda}_{12}=0$. In the next subsection, we consider the first case, isosceles $LRE$ (intersection of $\sigma_{1}=\sigma_{2}$ and $\lambda_{23}=0$). The subsection 4.1.2 is devoted to the bifurcation of isosceles $LRE$ and isosceles $ERE$. The second case, ($\tilde{\lambda}_{12}=0$ and $\lambda_{23}=0$) describes scalene $LRE$. In subsection 4.2, we will treat the bifurcation of isosceles $LRE$ and scalene $LRE$ ($\sigma_{1}=\sigma_{2}$ and $\tilde{\lambda}_{12}=0$ and $\lambda_{23}=0$).