Homographic solutions of the curved 3-body problem

# Homographic solutions of the curved 3-body problem

## Abstract.

In the 2-dimensional curved -body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic Eulerian solutions are the hyperbolic Eulerian relative equilibria, a result that proves their instability.

Florin Diacu

Pacific Institute for the Mathematical Sciences

and

Department of Mathematics and Statistics

University of Victoria

P.O. Box 3060 STN CSC

diacu@math.uvic.ca

and

Ernesto Pérez-Chavela

Departamento de Matemáticas

Apdo. 55534, México, D.F., México

epc@xanum.uam.mx

August 9, 2019

## 1. Introduction

We consider the -body problem in spaces of constant curvature (), which we will call the curved -body problem, to distinguish it from its classical Euclidean () analogue. The study of this problem might help us understand the nature of the physical space. Gauss allegedly tried to determine the nature of space by measuring the angles of a triangle formed by the peaks of three mountains. Even if the goal of his topographic measurements was different from what anecdotical history attributes to him (see [6]), this method of deciding the nature of space remains valid for astronomical distances. But since we cannot measure the angles of cosmic triangles, we could alternatively check whether specific (potentially observable) motions of celestial bodies occur in spaces of negative, zero, or positive curvature, respectively.

In [2], we showed that while Lagrangian orbits (rotating equilateral triangles having the bodies at their vertices) of non-equal masses are known to occur for , they must have equal masses for . Since Lagrangian solutions of non-equal masses exist in our solar system (for example, the triangle formed by the Sun, Jupiter, and the Trojan asteroids), we can conclude that, if assumed to have constant curvature, the physical space is Euclidean for distances of the order AU. The discovery of new orbits of the curved -body problem, as defined here in the spirit of an old tradition, might help us extend our understanding of space to larger scales.

This tradition started in the 1830s, when Bolyai and Lobachevsky proposed a curved 2-body problem, which was broadly studied (see most of the 77 references in [2]). But until recently nobody extended the problem beyond two bodies. The newest results occur in [2], a paper in which we obtained a unified framework that offers the equations of motion of the curved -body problem for any and . We also proved the existence of several classes of relative equilibria, including the Lagrangian orbits mentioned above. Relative equilibria are orbits for which the configuration of the system remains congruent with itself for all time, i.e. the distances between any two bodies are constant during the motion.

So far, the only other existing paper on the curved -body problem, treated in a unified context, deals with singularities, [3], a subject we will not approach here. But relative equilibria can be put in a broader perspective. They are also the object of Saari’s conjecture (see [7], [4]), which we partially solved for the curved -body problem, [2]. Saari’s conjecture has recently generated a lot of interest in classical celestial mechanics (see the references in [4], [5]) and is still unsolved for . Moreover, it led to the formulation of Saari’s homographic conjecture, [7], [5], a problem that is directly related to the purpose of this research.

We study here certain solutions that are more general than relative equilibria, namely orbits for which the configuration of the system remains similar with itself. In this class of solutions, the relative distances between particles may change proportionally during the motion, i.e. the size of the system could vary, though its shape remains the same. We will call these solutions homographic, in agreement with the classical terminology, [8].

In the classical Newtonian case, [8], as well as in more general classical contexts, [1], the standard concept for understanding homographic solutions is that of central configuration. This notion, however, seems to have no meaningful analogue in spaces of constant curvature, therefore we had to come up with a new approach.

Unlike in Euclidean space, homographic orbits are not planar, unless they are relative equilibria. In the case , for instance, the intersection between a plane and a sphere is a circle, but the configuration of a solution confined to a circle cannot expand or contract and remain similar to itself. Therefore the study of homographic solutions that are not relative equilibria is apparently more complicated than in the classical case, in which all homographic orbits are planar.

We focus here on three types of homographic solutions. The first, which we call Lagrangian, form an equilateral triangle at every time instant. We ask that the plane of this triangle be always orthogonal to the rotation axis. This assumption seems to be natural because, as proved in [2], Lagrangian relative equilibria, which are particular homographic Lagrangian orbits, obey this property. We prove the existence of homographic Lagrangian orbits in Section 3, and provide their complete classification in the case of equal masses in Section 4, for , and Section 5, for . Moreover, we show in Section 6 that Lagrangian solutions with non-equal masses don’t exist.

We then study another type of homographic solutions of the curved -body problem, which we call Eulerian, in analogy with the classical case that refers to bodies confined to a rotating straight line. At every time instant, the bodies of an Eulerian homographic orbit are on a (possibly) rotating geodesic. In Section 7 we prove the existence of these orbits. Moreover, for equal masses, we provide their complete classification in Section 8, for , and Section 9, for .

Finally, in Section 10, we discuss the existence of hyperbolic homographic solutions, which occur only for negative curvature. We prove that when the bodies are on the same hyperbolically rotating geodesic, a class of solutions we call hyperbolic Eulerian, every orbit is a hyperbolic Eulerian relative equilibrium. Therefore hyperbolic Eulerian relative equilibria are unstable, a fact that makes them unlikely observable candidates in a (hypothetically) hyperbolic physical universe.

## 2. Equations of motion

We consider the equations of motion on -dimensional manifolds of constant curvature, namely spheres embedded in , for , and hyperboloids1 embedded in the Minkovski space , for .

Consider the masses in , for , and in , for , whose positions are given by the vectors . Let be the configuration of the system, and , with , representing the momentum. We define the gradient operator with respect to the vector as

 ˜∇qi=(∂xi,∂yi,σ∂zi),

where is the signature function,

 (1) σ={+1,  for  κ>0−1,  for  κ<0,

and let denote the operator . For the 3-dimensional vectors and , we define the inner product

 (2) a⊙b:=(axbx+ayby+σazbz)

and the cross product

 (3) a⊗b:=(aybz−azby,azbx−axbz,σ(axby−aybx)).

The Hamiltonian function of the system describing the motion of the -body problem in spaces of constant curvature is

 Hκ(q,p)=Tκ(q,p)−Uκ(q),

where

 Tκ(q,p)=123∑i=1m−1i(pi⊙pi)(κqi⊙qi)

defines the kinetic energy and

 (4) Uκ(q)=∑1≤i

is the force function, representing the potential energy2. Then the Hamiltonian form of the equations of motion is given by the system

 (5) {˙qi=m−1ipi,˙pi=˜∇qiUκ(q)−m−1iκ(pi⊙pi)qi,  i=1,2,3, κ≠0,

where the gradient of the force function has the expression

 (6) ˜∇qiUκ(q)=3∑j=1j≠imimj|κ|3/2(κqj⊙qj)[(κqi⊙qi)qj−(κqi⊙qj)qi][σ(κqi⊙qi)(κqj⊙qj)−σ(κqi⊙qj)2]3/2.

The motion is confined to the surface of nonzero constant curvature , i.e. , where is the cotangent bundle of the configuration space , and

 M2κ={(x,y,z)∈R3 | κ(x2+y2+σz2)=1}.

In particular, is the 2-dimensional sphere, and is the 2-dimensional hyperbolic plane, represented by the upper sheet of the hyperboloid of two sheets (see the Appendix of [2] for more details). We will also denote by for , and by for .

Notice that the constraints given by imply that , so the -dimensional system (5) has constraints. The Hamiltonian function provides the integral of energy,

 Hκ(q,p)=h,

where is the energy constant. Equations (5) also have the integrals of the angular momentum,

 (7) 3∑i=1qi⊗pi=c,

where is a constant vector. Unlike in the Euclidean case, there are no integrals of the center of mass and linear momentum. Their absence complicates the study of the problem since many of the standard methods don’t apply anymore.

Using the fact that for , we can write system (5) as

 (8) ¨qi=3∑j=1j≠imj|κ|3/2[qj−(κqi⊙qj)qi][σ−σ(κqi⊙qj)2]3/2−(κ˙qi⊙˙qi)qi,  i=1,2,3,

which is the form of the equations of motion we will use in this paper.

## 3. Local existence and uniqueness of Lagrangian solutions

In this section we define the Lagrangian solutions of the curved 3-body problem, which form a particular class of homographic orbits. Then, for equal masses and suitable initial conditions, we prove their local existence and uniqueness.

###### Definition 1.

A solution of equations (8) is called Lagrangian if, at every time , the masses form an equilateral triangle that is orthogonal to the axis.

According to Definition 1, the size of a Lagrangian solution can vary, but its shape is always the same. Moreover, all masses have the same coordinate , which may also vary in time, though the triangle is always perpendicular to the axis.

We can represent a Lagrangian solution of the curved 3-body problem in the form

 (9) q=(q1,q2,q3),  with  qi=(xi,yi,zi), i=1,2,3,
 x1 =rcosω, y1 =rsinω, z1 =z, x2 =rcos(ω+2π/3), y2 =rsin(ω+2π/3), z2 =z, x3 =rcos(ω+4π/3), y3 =rsin(ω+4π/3), z3 =z,

where satisfies ; is the signature function defined in (1); is the size function; and is the angular function.

Indeed, for every time , we have that , which means that the bodies stay on the surface , each body has the same coordinate, i.e. the plane of the triangle is orthogonal to the axis, and the angles between any two bodies, seen from the geometric center of the triangle, are always the same, so the triangle remains equilateral. Therefore representation (9) of the Lagrangian orbits agrees with Definition 1.

###### Definition 2.

A Lagrangian solution of equations (8) is called Lagrangian homothetic if the equilateral triangle expands or contracts, but does not rotate around the axis.

In terms of representation (9), a Lagrangian solution is Lagrangian homothetic if is constant, but is not constant. Such orbits occur, for instance, when three bodies of equal masses lying initially in the same open hemisphere are released with zero velocities from an equilateral configuration, to end up in a triple collision.

###### Definition 3.

A Lagrangian solution of equations (8) is called a Lagrangian relative equilibrium if the triangle rotates around the axis without expanding or contracting.

In terms of representation (9), a Lagrangian relative equilibrium occurs when is constant, but is not constant. Of course, Lagrangian homothetic solutions and Lagrangian relative equilibria, whose existence we proved in [2], are particular Lagrangian orbits, but we expect that the Lagrangian orbits are not reduced to them. We now show this by proving the local existence and uniqueness of Lagrangian solutions that are neither Lagrangian homothetic, nor Lagrangian relative equilibria.

###### Theorem 1.

In the curved -body problem of equal masses, for every set of initial conditions belonging to a certain class, the local existence and uniqueness of a Lagrangian solution, which is neither Lagrangian homothetic nor a Lagrangian relative equilibrium, is assured.

###### Proof.

We will check to see if equations (8) admit solutions of the form (9) that start in the region and for which both and are not constant. We compute then that

 κqi⊙qj=1−3κr2/2  for  i,j=1,2,3,  with  i≠j,
 ˙x1 =˙rcosω−r˙ωsinω, ˙y1 =˙rsinω+r˙ωcosω,
 ˙x2=˙rcos(ω+2π3)−r˙ωsin(ω+2π3),
 ˙y2=˙rsin(ω+2π3)+r˙ωcos(ω+2π3),
 ˙x3=˙rcos(ω+4π3)−r˙ωsin(ω+4π3),
 ˙y3=˙rsin(ω+4π3)+r˙ωcos(ω+4π3),
 (10) ˙z1=˙z2=˙z3=−σr˙r(σκ−1−σr2)−1/2,
 κ˙qi⊙˙qi=κr2˙ω2+κ˙r21−κr2  for  i=1,2,3,
 ¨x1=(¨r−r˙ω2)cosω−(r¨ω+2˙r˙ω)sinω,
 ¨y1=(¨r−r˙ω2)sinω+(r¨ω+2˙r˙ω)cosω,
 ¨z1=¨z2=¨z3=−σr¨r(σκ−1−σr2)−1/2−κ−1˙r2(σκ−1−σr2)−3/2.

Substituting these expressions into system (8), we are led to the system below, where the double-dot terms on the left indicate to which differential equation each algebraic equation corresponds:

 ¨x1: Acosω−Bsinω=0, ¨x2: Acos(ω+2π3)−Bsin(ω+2π3)=0, ¨x3: Acos(ω+4π3)−Bsin(ω+4π3)=0, ¨y1: Asinω+Bcosω=0, ¨y2: Asin(ω+2π3)+Bcos(ω+2π3)=0, ¨y3: Asin(ω+4π3)+Bcos(ω+4π3)=0, ¨z1,¨z2,¨z3: A=0,

where

 A:=A(t)=¨r−r(1−κr2)˙ω2+κr˙r21−κr2+24m(1−κr2)r2(12−9κr2)3/2,
 B:=B(t)=r¨ω+2˙r˙ω.

Obviously, the above system has solutions if and only if , which means that the local existence and uniqueness of Lagrangian orbits with equal masses is equivalent to the existence of solutions of the system of differential equations

 (11) ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩˙r=ν˙w=−2νwr˙ν=r(1−κr2)w2−κrν21−κr2−24m(1−κr2)r2(12−9κr2)3/2,

with initial conditions where . The functions , and are analytic, and as long as the initial conditions satisfy the conditions for all , as well as for , standard results of the theory of differential equations guarantee the local existence and uniqueness of a solution of equations (11), and therefore the local existence and uniqueness of a Lagrangian orbit with and not constant. The proof is now complete. ∎

## 4. Classification of Lagrangian solutions for κ>0

We can now state and prove the following result:

###### Theorem 2.

In the curved -body problem with equal masses and there are five classes of Lagrangian solutions:

(i) Lagrangian homothetic orbits that begin or end in total collision in finite time;

(ii) Lagrangian relative equilibria that move on a circle;

(iii) Lagrangian periodic orbits that are neither Lagrangian homothetic nor Lagrangian relative equilibria;

(iv) Lagrangian non-periodic, non-collision orbits that eject at time , with zero velocity, from the equator, reach a maximum distance from the equator, which depends on the initial conditions, and return to the equator, with zero velocity, at time .

None of the above orbits can cross the equator, defined as the great circle of the sphere orthogonal to the axis.

(v) Lagrangian equilibrium points, when the three equal masses are fixed on the equator at the vertices of an equilateral triangle.

The rest of this section is dedicated to the proof of this theorem.

Let us start by noticing that the first two equations of system (11) imply that , which leads to

 w=cr2,

where is a constant. The case can occur only when , which means . Under these circumstances the angular velocity is zero, so the motion is homothetic. These are the orbits whose existence is stated in Theorem 2 (i). They occur only when the angular momentum is zero, and lead to a triple collision in the future or in the past, depending on the sense of the velocity vectors.

For the rest of this section, we assume that . Then system (11) takes the form

 (12) ⎧⎨⎩˙r=ν˙ν=c2(1−κr2)r3−κrν21−κr2−24m(1−κr2)r2(12−9κr2)3/2.

Notice that the term of the last equation arises from the derivatives in (10). But these derivatives would be zero if the equilateral triangle rotates along the equator, because is constant in this case, so the term vanishes. Therefore the existence of equilateral relative equilibria on the equator (included in statement (ii) above), and the existence of equilibrium points (stated in (v))—results proved in [2]—are in agreement with the above equations. Nevertheless, the term stops any orbit from crossing the equator, a fact mentioned before statement (v) of Theorem 2.

Understanding system (12) is the key to proving Theorem 2. We start with the following facts:

###### Lemma 1.

Assume and . Then for , system (12) has two fixed points, while for it has one fixed point.

###### Proof.

The fixed points of system (12) are given by Substituting in the second equation of (12), we obtain

 1−κr2r2[c2r−24m(12−9κr2)3/2]=0.

The above remarks show that, for , is a fixed point, which physically represents an equilateral relative equilibrium moving along the equator. Other potential fixed points of system (12) are given by the equation

 c2(12−9κr2)3/2=24mr,

whose solutions are the roots of the polynomial

 (13) 729c4κ3r6−2916c4κ2r4+144(27c4κ+4m2)r2−1728.

Writing and assuming , this polynomial takes the form

 (14) p(x)=729κ3x3−2916c4κ2x2+144(27c4κ+4m2)x−1728,

and its derivative is given by

 (15) p′(x)=2187c4κ3x2−5832c4κ2x+144(27c4κ+4m2).

The discriminant of is

By Descartes’s rule of signs, can have one or three positive roots. If has three positive roots, then must have two positive roots, but this is not possible because its discriminant is negative. Consequently has exactly one positive root.

For the point to be a fixed point of equations (12), must satify the inequalities . If we denote

 (16) g(r)=c2r−24m(12−9κr2)3/2,

we see that, for , is a decreasing function since

 (17) ddrg(r)=−c2r2−648mκr(12−9κr2)5/2<0.

When , we obviously have that since we assumed . When , we have . If , then , so is not a fixed point. Therefore, assuming , a necessary condition that is a fixed point of system (12) with is that

 κ1/2c2−(8/√3)m<0.

For the only fixed point of system (12) is . This conclusion completes the proof of the lemma. ∎

### 4.1. The flow in the (r,ν) plane for κ>0

We will now study the flow of system (12) in the plane for . At every point with , the slope of the vector field is given by , i.e. by the ratio where

 h(r,ν)=c2(1−κr2)νr3−κrν1−κr2−24m(1−κr2)νr2(12−9κr2)3/2.

Since , the flow of system (12) is symmetric with respect to the axis for . Also notice that, except for the fixed point , system (12) is undefined on the lines and . Therefore the flow of system (12) exists only for points in the band and for the point .

Since , no interval on the axis can be an invariant set for system (12). Then the symmetry of the flow relative to the axis implies that orbits cross the axis perpendicularly. But since at every non-fixed point, the flow crosses the axis perpendicularly everywhere, except at the fixed points.

Let us further treat the case of one fixed point and the case of two fixed points separately.

#### The case of one fixed point

A single fixed point, namely , appears when . Then the function , which is decreasing, has no zeroes for , therefore in this interval, so the flow always crosses the axis upwards.

For , the right hand side of the second equation of (12) can be written as

 (18) G(r,ν)=g1(r)g(r)+g2(r,ν),

where

 (19) g1(r)=1−κr2r2  and  g2(r,ν)=−κrν21−κr2.

But and So, like , the functions and are decreasing in , with , therefore is a decreasing function as well. Consequently, for , the slope of the vector field decreases from at to ar . For , the slope of the vector field increases from at to at .

This behavior of the vector field forces every orbit to eject downwards from the fixed point, at time and with zero velocity, on a trajectory tangent to the line , reach slope zero at some moment in time, then cross the axis perpendicularly upwards and symmetrically return with final zero velocity, at time , to the fixed point (see Figure 1(a)). So the flow of system (12) consists in this case solely of homoclinic orbits to the fixed point , orbits whose existence is claimed in Theorem 2 (iv). Some of these trajectories may come very close to a total collapse, which they will never reach because only solutions with zero angular momentum (like the homothetic orbits) encounter total collisions, as proved in [3].

So the orbits cannot reach any singularity of the line , and neither can they begin or end in a singularity of the line . The reason for the latter is that such points are or the form with , therefore at such points. But the vector field tends to infinity when approaching the line , so the flow must be tangent to it, consequently must tend to zero, which is a contradiction. Therefore only homoclinic orbits exist in this case.

#### The case of two fixed points

Two fixed points, and , with , occur when . Since is decreasing in the interval , we can conclude that for and for . Therefore the flow of system (12) crosses the axis upwards when , but downwards for (see Figure 1(b)).

The function , defined in (18), fails to be decreasing in the interval along lines of constant , but it has no singularities in this interval and still maintains the properties

 limr→0+G(r,ν)=+∞  and limr→(κ−1/2)−G(r,ν)=−∞.

Therefore must vanish at some point, so due to the symmetry of the vector field with respect to the axis, the fixed point is surrounded by periodic orbits. The points where vanishes are given by the nullcline , which has the expression

 ν2=(1−κr2)2κr3[c2r−24m(12−9κr2)3/2].

This nullcline is a disconnected set, formed by the fixed point and a continuous curve, symmetric with respect to the axis. Indeed, since the equation of the nullcline can be written as , and in the case of two fixed points (as shown in the proof of Lemma 1), only the point satisfies the nullcline equation away from the fixed point .

The asymptotic behavior of near also forces the flow to produce homoclinic orbits for the fixed point , as in the case discussed in Subsection 4.1.1. The existence of these two kinds of solutions is stated in Theorem 2 (iii) and (iv), respectively. The fact that orbits cannot begin or end at any of the singularities of the lines or follows as in Subsection 4.1.1. This remark completes the proof of Theorem 2.

## 5. Classification of Lagrangian solutions for κ<0

We can now state and prove the following result:

###### Theorem 3.

In the curved -body problem with equal masses and there are eight classes of Lagrangian solutions:

(i) Lagrangian homothetic orbits that begin or end in total collision in finite time;

(ii) Lagrangian relative equilibria, for which the bodies move on a circle parallel with the plane;

(iii) Lagrangian periodic orbits that are not Lagrangian relative equilibria;

(iv) Lagrangian orbits that eject at time from a certain relative equilibrium solution (whose existence and position depend on the values of the parameters) and returns to it at time ;

(v) Lagrangian orbits that come from infinity at time and reach the relative equilibrium at time ;

(vi) Lagrangian orbits that eject from the relative equilibrium at time and reach infinity at time ;

(vii) Lagrangian orbits that come from infinity at time and symmetrically return to infinity at time , never able to reach the Lagrangian relative equilibrium ;

(viii) Lagrangian orbits that come from infinity at time , reach a position close to a total collision, and symmetrically return to infinity at time .

The rest of this section is dedicated to the proof of this theorem. Notice first that the orbits described in Theorem 3 (i) occur for zero angular momentum, when , as for instance when the three equal masses are released with zero velocities from the Lagrangian configuration, a case in which a total collapse takes place at the point . Depending on the initial conditions, the motion can be bounded or unbounded. The existence of the orbits described in Theorem 3 (ii) was proved in [2]. To address the other points of Theorem 3, and show that no other orbits than the ones stated there exist, we need to study the flow of system (12) for . Let us first prove the following fact:

###### Lemma 2.

Assume , and . Then system (12) has no fixed points when , and can have two, one, or no fixed points when .

###### Proof.

The number of fixed points of system (12) is the same as the number of positive zeroes of the polynomial defined in (14). If , all coefficients of are negative, so by Descartes’s rule of signs, has no positive roots.

Now assume that . Then the zeroes of are the same as the zeroes of the monic polynomial (i.e. with leading coefficient 1):

 ¯p(x)=x3−4κ−1x2+[48κ−2+(64/81)c−4κ−3m2]x−(64/27)κ−3,

obtained when dividing by the leading coefficient. But a monic cubic polynomial can be written as

 x3−(a1+a2+a3)x2+(a1a2+a2a3+a3a1)x−a1a2a3,

where and are its roots. One of these roots is always real and has the opposite sign of . Since the free term of is positive, one of its roots is always negative, independently of the allowed values of the coefficients . Consequently can have two positive roots (including the possibility of a double positive root) or no positive root at all. Therefore system (12) can have two, one, or no fixed points. As we will see later, all three cases occur. ∎

We further state and prove a property, which we will use to establish Lemma 4:

###### Lemma 3.

Assume , let be a fixed point of system (12), and consider the function defined in (16). Then if and only if . Moreover, .

###### Proof.

Since is a fixed point of system (12), it follows that . Then it follows from relation (16) that . Substituting this value of into the equation , which is equivalent to

 648mκr∗(12−9κr2∗)5/2=−c2r2∗,

it follows that . Therefore . Obviously, for this value of , , so the first part of Lemma 3 is proved. To prove the second part, substitute into the equation , which is then equivalent with the relation

 (20) 9√3c2(−κ)1/2−4m=0.

Notice that

 d2dr2g(r)=2c2r3−648mκ(12−9κr2)5/2−29160mκ2r2(12−9κr2)7/2.

Substituting for in the above equation, and using (20), we are led to the conclusion that , which is positive for . This completes the proof. ∎

The following result is important for understanding a qualitative aspect of the flow of system (12), which we will discuss later in this section.

###### Lemma 4.

Assume , and let be a fixed point of system (12). If , then , where is defined in (18).

###### Proof.

Since is a fixed point of (12), . But for , we have , so necessarily . Moreover, and since , it follows that . But

 ∂G∂r(r,ν)=ddrg1(r)⋅g(r)+g1(r)ddrg(r)+∂∂rg2(r,ν),

so the condition implies that . By Lemma 3, and . Using now the fact that

 ∂2G∂r2(r,ν)=d2dr2g1(r)g(r)+2ddrg1(r)ddrg(r)+g1(r)d2dr2g(r)+∂2∂r2g2(r,ν),

it follows that Since Lemma 3 implies that , and we know that , it follows that , a conclusion that completes the proof. ∎

### 5.1. The flow in the (r,ν) plane for κ<0

We will now study the flow of system (12) in the plane for . As in the case , and for the same reason, the flow is symmetric with respect to the axis, which it crosses perpendicularly at every non-fixed point with . Since we can have two, one, or no fixed points, we will treat each case separately.

#### The case of no fixed points

No fixed points occur when has no zeroes. Since