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Reduction and relative equilibria for the 2-body
problem in spaces of constant curvature
A.V. Borisov, L.C. Garc´ıa-Naranjo, I.S. Mamaev & J. Montaldi
August 15, 2017
Abstract
We perform the reduction of the two-body problem in the two dimensional spaces
of constant non-zero curvature and we use the reduced equations of motion to classify
all relative equilibria (RE) of the problem and to study their stability. In the negative
curvature case, the nonlinear stability of the stable RE is established by using the reduced
Hamiltonian as a Lyapunov function. Surprisingly, in the positive curvature case this
approach is not sufficient and the proof of nonlinear stability requires the calculation of
Birkhoff normal forms and the application of stability methods coming from KAM theory.
In both cases we summarize our results in the Energy-Momentum bifurcation diagram of
the system.
Introduction 2
1 Reduction in the case S24
1.1 Group parameterization of configurations .................... 4
1.2 Integrals of motion and reduction ......................... 5
1.3 Reconstruction ................................... 7
1.4 Example — the 2-body problem on S2...................... 7
2 Reduction in the case L28
2.1 Group parameterization of configurations .................... 8
2.2 The 2-body problem ................................ 9
3 Relative equilibria for the 2-body problem on the Lobachevsky plane 11
3.1 Existence and classification of relative equilibria ................ 11
3.2 Stability analysis of the relative equilibria .................... 13
3.3 Topology of the Energy-Momentum level surfaces ................ 16
4 Relative equilibria for the 2-body problem on the sphere 17
4.1 The case of different masses ............................ 18
4.2 The case of equal masses .............................. 20
4.3 The signature of the Hessian of the reduced Hamiltonian along the relative
equilibria .................................... 20
4.4 Stability of acute relative equilibria in the case of different masses ...... 23
Open problems 27
1
Introduction
The study of the dynamics of material points and rigid bodies in spaces of constant
curvature has been a popular subject of research in the past two decades. For recent
advances in this area, see the review [4,6] and the book by Diacu [11].
In this paper we focus on the 2-body problem on a (complete and simply connected)
two dimensional space of constant non-zero curvature. Contrary to the situation in flat
space, the system is not equivalent to the corresponding generalisation of the Kepler
problem: it is nonintegrable and exhibits chaotic behaviour. Recent papers like [2,6,7,
13,14,20,21] have considered the reduction by symmetries and some qualitative aspects
of the problem. Libration points and choreographies are treated in [7,14,16] and the
restricted two-body problem is considered in [5,10].
Our main contributions consist of using the explicit form of the reduced system to
classify all relative equilibria (RE) of the problem and to study their stability. We describe
this in more detail below.
Reduction
As mentioned above, the reduction of the problem has been considered before [2,7]. We
have nevertheless included a self-contained presentation of the reduction for completeness.
For both the positive and negative curvature cases, the unreduced system is a four
degree of freedom symplectic Hamiltonian system, the symmetry group is three dimen-
sional and acts freely and properly. The reduced system is a five dimensional Poisson
Hamiltonian system, whose generic symplectic leaves are the four dimensional level sets
of a Casimir function.
We first deal with the case of positive curvature and consider the general N-body
problem on the 2-dimensional sphere S2. We perform the reduction of the problem by the
action of SO(3) that simultaneously rotates all of the masses. We proceed by introducing a
moving coordinate frame whose axes are aligned according to the configuration of the first
two bodies in a convenient way. The Hamiltonian of the system may then be written in
terms of generalised coordinates for the positions of the masses with respect to the moving
frame, their corresponding generalised momenta, and the vector of angular momentum
mwritten in the moving frame. These quantities do not depend on the orientation of the
fixed frame and may therefore be used as coordinates on the reduced space. This approach
is inspired by the reduction of the free rigid body problem and, just as it happens for
that problem, the Euclidean squared norm of mpasses down to the quotient space as a
Casimir function whose level sets are the symplectic leaves of the reduced space.
We apply an analogous reduction scheme in the case of negative curvature by consid-
ering the action of SO(2,1) on the pseudo-sphere L2. This time, the Casimir function C
on the reduced space is the squared norm of the momentum vector mwith respect to the
Minkowski metric.
Classification and stability of relative equilibria
We classify the RE of the problem by finding all of the extrema of the reduced Hamiltonian
restricted to the the level sets of the corresponding Casimir function. In this way we
recover the results of [8,12,13] in a systematic and elementary fashion. Moreover, in this
manner, we arrive at a convenient position to analyse their stability.
The case of negative curvature
As it is known [12,13], there are two families of RE, hyperbolic and elliptic. The former
are unbounded solutions that do not have an analog in euclidean space, while the latter are
2
periodic solutions that generalise the RE in euclidean space. This classification becomes
very transparent in our treatment: hyperbolic RE correspond to extrema of the reduced
Hamiltonian restricted to negative values of the Casimir function C, whereas elliptic RE
correspond to positive ones.
The (nonlinear) stability properties of these RE was first established in [13] by working
on a symplectic slice for the unreduced system since the reduced equations were not
available in a journal in English. With the reduced system at hand, we are able to recover
these results in elementary terms by directly analysing the signature of the Hessian of
the reduced Hamiltonian at the RE. Hyperbolic RE are always unstable, whereas elliptic
RE are stable if the masses are sufficiently close. However, as the distance between
them grows, the family undergoes a saddle-node bifurcation and the elliptic RE become
unstable.
All of the information of the RE of the system is conveniently illustrated in the Energy-
Momentum bifurcation diagram 4, that is presented here for the first time. This kind of
diagram, as well as the underlying topological considerations of the analysis, goes back to
Smale and has been been developed in detail for integrable systems by Bolsinov, Borisov
and Mamaev [8]. The application of this kind of analysis to nonintegrable systems, as the
one considered in this paper, is not very common.1
The case of positive curvature
In this case all RE are periodic solutions in which the masses rotate about an axis that
passes through the shortest geodesic that joins them. As indicated first in [7], the classi-
fication of RE in this case is more intricate than for negative curvature since it depends
on how the masses of the bodies compare to each other:
(i) If the masses are different, there are two disjoint families of RE that we term acute
and obtuse, according to the (constant) value of the angle between the masses along
the motion. For acute RE, it is the heavier mass that is closer to the axis of rotation
and hence these are a natural generalisation of the RE of the problem in euclidean
space. On the other hand, for obtuse RE it is the lighter mass which is closer to the
axis of rotation, and these RE to not have an analog in euclidean (or hyperbolic)
space.
(ii) If the masses are equal, there are two families of RE. We call Isosceles RE those for
which the axis of rotation bisects the arc that joins the two masses, and right angle
RE those for which the angle between the masses is π/2 and the axis of rotation
is located anywhere between them. These two families intersect when the axis of
rotation of right angle RE bisects the arc between the masses, and a bifurcation of
the RE of the system takes place.
As for the case of negative curvature, we compute the signature of the Hessian of
the reduced Hamiltonian at the RE in an attempt to establish stability results. Via this
analysis it is possible to conclude the instability of certain RE of the system. However,
contrary to the case of negative curvature, this is insufficient to prove any kind of nonlinear
stability results of the RE since the Hessian matrix is not definite and hence the reduced
Hamiltonian may not be used as a Lyapunov function of the system. This surprising
feature of the problem was also found in [18] by working on a symplectic slice of the
unreduced system.
In view of the above considerations, we took an analytical approach to the study of
the nonlinear stability of certain RE of the problem. By using Birkhoff normal forms
and applying KAM theory, we are able to show that, in the case of different masses, the
generic acute RE of the problem are stable.
1We only mention the paper [9] on the Conley index (where new isosceles vortex configurations were found
and their stability was established using topological methods).
3
Outline of the paper
The reduction of the problem on S2and L2is respectively presented in Sections 1 and
2. The case of the positive curvature is presented first since it is more natural. We then
proceed to classify the RE of the problem and study their stability. We first deal with
the case of negative curvature in section 3 and then with the case of positive curvature
in section 4. We have chosen to present first the negative curvature results since, as was
discussed above, the analysis is more straightforward. Finally, some related open problems
are described at the end.
1 Reduction in the case S2
1.1 Group parameterization of configurations
Consider the N-body problem on a two-dimensional sphere S2⊂R3. Let OXY Z be a
fixed coordinate system and let Rα= (Xα, Yα, Zα) be the Cartesian coordinates of the
point mass µα,α= 1, . . . , N .
Let us choose a pair of particles µ1,µ2and suppose that a moving orthogonal coordi-
nate system Oxyz is attached to them in such a way that the axis Oz passes through the
point µ1and the plane Oyz contains both masses µ1,µ2(see Fig. 1). In this case, the
radius vectors of the point masses rα= (xα, yα, zα) in the moving axes Oxyz characterize
the relative position of the particles (i.e., the configuration of the bodies irrelative to its
position on the sphere S2). Let the corresponding generalized (local) coordinates, which
completely parameterize the relative position (configuration) of the particles, be denoted
by q= (q1, . . . , qn) in the case of Nmaterial points n= 2N−3. We will describe the
orientation of the moving axes relative to the fixed axes by the Euler angles θ,ϕ,ψso
that the position of the particles in the fixed axes is described by
Rα(θ, ϕ, ψ, q) = Q(θ , ϕ, ψ)·rα(q),
Q=QψQθQϕ=
cos ϕcos ψ−cos θsin ψsin ϕ−sin ϕcos ψ−cos θsin ψcos ϕsin θsin ψ
cos ϕsin ψ+ cos θcos ψsin ϕ−sin ϕsin ψ+ cos θcos ψcos ϕ−sin θcos ψ
sin θsin ϕsin θcos ϕcos θ
,
Qθ=
1 0 0
0 cos θ−sin θ
0 sin θcos θ
,Qϕ=
cos ϕ−sin ϕ0
sin ϕcos ϕ0
0 0 1
,Qψ=
cos ψ−sin ψ0
sin ψcos ψ0
0 0 1
,
(1)
where Qis the matrix of the direction cosines.
Z
Y
q
Figure 1: Euler angles for the 2-body configuration on S2.
4
Assuming that the forces of interaction are potential, we construct the Lagrangian of
the system
L=T−U,
where Tand Uare the kinetic and potential energy, respectively. We use a relation
(well-known in rigid body dynamics) for the angular velocity matrix:
ˆ
ω=Q−1˙
Q=
0−ωzωy
ωz0−ωx
−ωyωx0
=
=
0−˙
ψcos θ−˙ϕ˙
ψsin θcos ϕ−˙
θsin ϕ
˙
ψcos θ+ ˙ϕ0−˙
ψsin θsin ϕ−˙
θcos ϕ
−˙
ψsin θcos ϕ+˙
θsin ϕ˙
ψsin θsin ϕ+˙
θcos ϕ0
,
(2)
where ω= (ωx, ωy, ωz) are the projections of the angular velocity of the moving frame
onto the moving axes Oxyz, and we represent the kinetic energy of the system as
T=X
α
µα˙
Rα,˙
Rα=1
2ω,I(q)ω+ω,ξ(q,˙
q)+
+1
2X
i,j
Gij (q) ˙qi˙qj,
I(q) = X
α
µαr2
αE−rα⊗rα,ξ=X
α
µαrα×˙
rα=X
α,i
µαrα×∂rα
∂qi
˙qi,
Gij =X
α
µα∂rα
∂qi
,∂rα
∂qj,
where q= (q1, . . . , qn), the notation (·,·), ×and ⊗corresponds to the scalar, vector and
tensor product in R3and Eis the identity matrix. Since the particles interact only with
each other, the potential energy of the system does not depend on the Euler angles:
U=U(q).
1.2 Integrals of motion and reduction
As is well known, due to the invariance of the Lagrangian under rotations (i.e., under the
change of fixed axes) the projections of the angular momentum of the system onto the
fixed axes are preserved:
M=X
α
µαRα×˙
Rα=QI(q)ω+ξ= const.
Using the Noether theorem, we obtain the corresponding expressions for the compo-
nents of this vector in the Euler angles:
MX= cos ψ∂L
∂˙
θ+sin ψ
sin θ∂L
∂˙ϕ−cos θ∂L
∂˙
ψ,
MY= sin ψ∂L
∂˙
θ−cos ψ
sin θ∂L
∂˙ϕ−cos θ∂L
∂˙
ψ,
MZ=∂L
∂˙
ψ.
(3)
5
We define the generalized momenta of the system in a standard way:
Pθ=∂L
∂˙
θ, Pϕ=∂L
∂˙ϕ, Pψ=∂L
∂˙
ψ,
pi=∂L
∂˙qi
, i = 1, . . . , n.
(4)
Then the equations of motion can be represented in the canonical Hamiltonian form:
˙
θ=∂H
∂Pθ
,˙
Pθ=−∂H
∂θ ,˙ϕ=∂H
∂Pϕ
,˙
Pϕ=−∂H
∂ϕ ,˙
ψ=∂H
∂Pψ
,˙
Pψ=−∂H
∂ψ ,
˙qi=∂H
∂pi
,˙pi=−∂ H
∂qi
.
(5)
The Hamiltonian function is expressed in a natural way in terms of the projections of
the angular momentum vector onto the moving axes m= (mx, my, mz):
H=1
2m,A(q)m+m,k(q,p)+1
2X
i,j
Cij (q)pipj+U(q),
mx=sin ϕ
sin θ(Pψ−Pϕcos θ) + Pθcos ϕ, my=cos ϕ
sin θ(Pψ−Pϕcos θ)−Pθsin ϕ,
mz=Pϕ,
k=B(q)p,
(6)
where A(q), B(q), C(q) are 3 ×3, 3 ×nand n×nmatrices which are the blocks of the
(3 + n)×(3 + n) matrix arising when the quadratic form corresponding to the kinetic
energy is inverted:
A B
BTC!=
I
∂ξi
∂˙qj
∂ξi
∂˙qj
T
G
−1
.
In order to obtain a reduced system, we pass from the canonical momenta Pθ,Pϕ,Pψ
to the variables mx,my,mz. It turns out that the set of variables m,q,pis closed
relative to the Poisson bracket:
{mi, mj}=−εijk mk,{qi, pj}=δij .
This Poisson bracket is degenerate and possesses the Casimir function
C0=m2
x+m2
y+m2
z.(7)
Since the Hamiltonian (6) is expressed only in terms of these variables, we obtain the
closed system of equations
˙
m=m×∂H
∂m,˙qi=∂H
∂pi
,˙pi=−∂ H
∂qi
,(8)
which defines the reduced system for this problem (since the variables m,q,pare invariant
under the left action of the group SO(3), i.e., under the change of the fixed axes).
As is well known [1], in order to obtain a reduced system in canonical variables, it is
necessary to define on the level set of the integral (7)
C0=M2
0
6
the cylindrical coordinates (Andoyer variables)
mx=p0, my=qM2
0−p2
0sin q0, mz=qM2
0−p2
0cos q0,
q0∈[0,2π), p0∈[−M0, M0],
which commute canonically:
{q0, p0}= 1.
1.3 Reconstruction
Assume that we are given a solution to the system (8)
m(t),q(t),p(t).
We need to determine the time dependence for the Euler angles.
1. As a first step, we choose the fixed axes OX Y Z in such a way that M||OZ. Hence,
the following relations hold:
MX= 0, MY= 0, MZ=Pψ=M0.
Using them, we find from (3) that
Pθ= 0, Pϕ−Pψcos θ=mz(t)−M0cos θ= 0.
Finally, taking into account (6), we find
Pθ= 0, Pϕ=mz(t), Pψ=M0= const,
which leads to the relations
mx=M0sin θsin ϕ, my=M0sin θcos ϕ, mz=M0cos θ. (9)
So finally we obtain
cos θ=mz(t)
M0
,tan ϕ=mx(t)
my(t).(10)
2. Using these relations, we obtain the quadrature for the angle ψ:
˙
ψ=sin ϕωx+ cos ϕωy
sin θ=M0(mx(t)ωx(t) + my(t)ωy(t))
M2
0−m2
z(t),
ωx=∂H
∂mx
, ωy=∂H
∂my
.
(11)
1.4 Example — the 2-body problem on S2
In this case, the system has only one mutual variable, which is the angle between the
radius vectors of the points (see Fig. 1)
q1∈(0, π).
If we denote the radius of the sphere by a, the radius vectors of the particles in the moving
coordinate system Oxyz are
r1= (0,0, a),r2= (0, a sin q1, a cos q1).(12)
7
Performing the above operations, we obtain the Hamiltonian of the system (6) in the form
H=1
2a2µ1(m,A(q1)m)+2mxp1+1 + µ1
µ2p2
1+U(q1),(13)
A(q1) =
1 0 0
0 1 B
0B C
,B=cos q1
sin q1
,C=µ1+µ2cos2q1
µ2sin2q1
.
Using (11) we obtain the quadrature for the angle of precession
µ1a2˙
ψ=M01 + mxp1+Bmymz
M2
0−m2
z.(14)
2 Reduction in the case L2
2.1 Group parameterization of configurations
Let us consider a three-dimensional Minkowski space and attach the fixed coordinate
system OXY Z to it. The scalar product is given by
hR,Rig= (R,gR),g = diag(1,1,−k2).(15)
Remark 2.1. If we formally substitute k=i, then we obtain a standard Euclidean scalar
product and all subsequent formulas will turn into the formulas of the previous section.
The pseudo-sphere L2(the Lobachevsky plane) in this case is represented as one of
the connectedness components of the two-sheet hyperboloid
hR,Rig=X2+Y2−k2Z2=−a2k2,(16)
where ais some real-valued parameter. For definiteness we set Z > 0. Choosing the local
coordinates on (16) in the form
X=ka sinh θcos ϕ, Y =ka sinh θsin ϕ, Z =acosh θ,
we obtain a standard expression for the metric (obtained under the restriction (15)):
ds2=a2k2(dθ2+ sinh2θdϕ2).
Its Gaussian curvature is constant and equal to
K=−1
a2k2.
We denote the coordinates of the particles in the fixed axes again as Rα= (Xα, Yα, Zα)
and their masses as µα,α= 1 . . . N .
As in the previous case, we choose a pair of particles µ1,µ2and attach to them a
moving coordinate system Oxyz which is orthogonal in the metric (15) and for which
both these points lie in the plane Oyz and the axis Oz passes through the point µ1. In
these moving axes the radius vectors of the point masses rα= (xα, yα, zα) define the
configuration of the bodies; they depend only on the internal generalized coordinates
q= (q1, . . . , qn).
As is well known, the transformation from the moving axes Oxy to the fixed axes
OXY Z is given by the matrix Q∈SO(2,1):
Rα=Qrα, α = 1 . . . N,
8
Z
Y
Figure 2: Generalized Euler angles for the 2-body configuration on L2.
which satisfies the equation
QTgQ =g.
To parameterize the group SO(2,1), we choose Qas a sequence of 3 rotations, one of
which is hyperbolic:
Qθ=
1 0 0
0 cosh θ k sinh θ
01
ksinh θcosh θ
,Qϕ=
cos ϕ−sin ϕ0
sin ϕcos ϕ0
0 0 1
,
Qψ=
cos ψ−sin ψ0
sin ψcos ψ0
0 0 1
,
Q=QψQθQϕ=
=
cos ϕcos ψ−cosh θsin ϕsin ψ−sin ϕcos ψ−cosh θcos ϕsin ψ−ksinh θsin ψ
cos ϕsin ψ+ cosh θsin ϕcos ψ−sin ϕsin ψ+ cosh θcos ϕcos ψ k sinh θcos ψ
1
ksinh θsin ϕ1
ksinh θcos ϕcosh θ
(to pass to SO(3), we have to set k=i,θ→iθ).
The matrix of the left-invariant angular velocities is given by the following relation,
analogous to (2):
b
ω=Q−1˙
Q=
0−ωz−k2ωy
ωz0k2ωx
−ωyωx0
=
=
0−˙ϕ−˙
ψcosh θ k(˙
θsin ϕ−˙
ψsinh θcos ϕ)
˙ϕ+˙
ψcosh θ0k(˙
θcos ϕ+˙
ψsinh θsin ϕ)
1
k(˙
θsin ϕ−˙
ψsinh θcos ϕ)1
k(˙
θcos ϕ+˙
ψsinh θsin ϕ) 0
.
2.2 The 2-body problem
Here we shall not develop a general construction for an arbitrary number of particles as
we did in the previous section, but consider in more detail the 2-body problem on L2.
In the chosen moving coordinate system Oxyz the position of the bodies is given by the
radius vectors
r1= (0,0, a), r2= (0, ak sinh q1, a cosh q1),
where q1∈R+= (0,∞) because in the case where a=k= 1 it corresponds to the
distance between the particles in L2.
9
In this case, the Lagrangian function is
L=T−U(q1),
where U(q1) is the potential energy of interaction and the kinetic energy Tcan be repre-
sented as
T=1
2X
α
µαh˙
Rα,˙
Rαig=1
2X
α
µαhˆ
ωrα+˙
rα,ˆ
ωrα+˙
rαig=
=a2k4
2µ1+µ2ω2
x+µ1+µ2cosh2q1ω2
y+µ2sinh2q1
k2ω2
z+
2µ2sinh q1cosh q1
kωyωz+µ2
kωx˙q1+µ2
k2˙q2
1.
The projections of the angular momentum vector to the axes attached to the body
and the momentum corresponding to the coordinate q1are given by
mx=∂L
∂ωx
=kPθcos ϕ−Pϕ
cosh θcos ϕ
sinh θ+Pψ
sin ϕ
sinh θ,
my=∂L
∂ωy
=k−Pθsin ϕ−Pϕ
cosh θcos ϕ
sinh θ+Pψ
cos ϕ
sinh θ,
mz=∂L
∂ωz
=Pϕ, p1=∂L
∂˙q1
.
The Poisson bracket of these functions has the form
{mx, my}=k2mz,{my, mz}=−mx,{mz, mx}=−my,(17)
{q1, p1}= 1.
The Hamiltonian of the system is defined by
H=X
i
miωi+p1˙q1−Lω→m,˙q1→p1
=
=1
2a2k4µ1m2
x+m2
y−2Bmymz+Cm2
z−2kmxp1+k21 + µ1
µ2p2
1+U(q1)
B=kcosh q1
sinh q1
,C=k2µ1+µ2cosh2q1
µ2sinh2q1
.
(18)
The equations of motion are written in vector form as follows:
˙
m= (gm)×∂H
∂m,˙q1=∂H
∂p1
,˙p1=−∂H
∂q1
.(19)
10
3 Relative equilibria for the 2-body problem on the
Lobachevsky plane
3.1 Existence and classification of relative equilibria
We denote p1=p,q1=q∈R+= (0,+∞). Let µ=µ1
µ2
be the quotient between the
masses. The reduced Hamiltonian (18) can be written as
H(m, q, p) = 1
2µ1a2(m,A(q)m)−2kmxp+k2(1 + µ)p2+U(q),(20)
where
A(q) =
1 0 0
0 1 −kcosh q
sinh q
0−kcosh q
sinh q
k2(µ+ cosh2q)
sinh2q
.
In our analysis we shall assume that the potential U(q) is purely attractive, meaning
that U0(q)>0. This situation is encountered in the gravitational case where
U(q) = −Gµ1µ2
tanh q,(21)
where G > 0 is proportional to the gravitational constant.
The Poisson bracket (17) has generic rank 4 (its rank drops to 2 when m= 0). Its
4-dimensional symplectic leaves are the regular level sets of the Casimir function
C(m) = −hm,mig=−m2
x−m2
y+k2m2
z,
that is a first integral of the reduced equations of motion (19).
The relative equilibria of the problem correspond to equilibrium points on the reduced
space. These are the local extrema of Hon the level sets of C.
Critical points of Hon the level sets of Care characterised as the solutions to the
following set of equations:
∂H
∂p = 0,(22a)
∂H
∂m×(gm) = 0,(22b)
∂H
∂q = 0,(22c)
where the matrix g is defined in (15) and ×is the vector product in R3. The condition
(22a) yields
p=mx
k(1 + µ).(23)
Substituting this expression into (22b) yields two possibilities:
1) my=mz= 0. In this case (22c) implies U0(q) = 0, and there is no solution for qby
our assumption that Uis strictly attractive.
2) mx= 0. We analyse this case in what follows assuming that myand mzdo not
vanish simultaneously, since otherwise we are back in case (i).
On the unreduced space, the qualitative properties of these solutions depend on the
sign that Ctakes along them:
11
(i) Relative equilibria having C > 0 are periodic solutions and are termed el liptic
relative equilibria. Their existence is due to a balance between centrifugal and
gravitational forces. These type of solutions also exist in the positive and zero
curvature cases.
(ii) Relative equilibria having C < 0 are unbounded solutions and are termed hyperbolic
relative equilibria. This type of relative equilibrium does not exist in the positive
or zero curvature case. As explained in [13] their existence is due to the property
of the Lovachevsky space that makes parallel geodesics “separate”. This separating
effect is balanced by the gravitational attraction in a very delicate manner. As we
shall see, these relative equilibria are all unstable.
(iii) Relative equilibria having C= 0 do not exist as we shall see below. If they did exist,
they would be called parabolic relative equilibria in a natural analogy with the
terminology for quadrics on the plane.
Existence and classification of elliptic relative equilibria. Recall that we have
assumed that mx= 0. We parametrise the open region of the reduced phase space having
C(m)>0 with the parameters M6= 0 and α∈R, by putting
my=Msinh α, mz=M
kcosh α. (24)
Then C(m) = M2and (22b) is satisfied provided that
sinh 2(q−α) = µsinh 2α. (25)
This necessary condition is equivalent to the one given in [13] by introducing the concept
of the hyperbolic centre of mass. Equation (25) admits the unique solution for α
α=q
2+1
4ln µ+e2q
1 + µe2q,(26)
With the above value of α, equation (22c) is satisfied provided that Mis such that
M2=µ1a2sinh3qU 0(q)
µcosh2αcosh q+ cosh αcosh(q−α).(27)
For an attractive potential, like the gravitational (20), the right hand side of this expression
is positive.
The choice of sign for Mcorresponds to two solutions related by reversing the direction
of time. We shall not distinguish these two. Therefore, we have shown the following.
Proposition 3.1. There exists a unique family of elliptic relative equilibria that is parametrised
by q. This family has mx=p= 0, and myand mzdefined by (24), where αand |M|are
determined by (26)and (27).
This family of relative equilibria corresponds to the bifurcation curve with a cusp on
the semi-plane C > 0 on the energy – momentum diagram (see Fig. 4).
Existence and classification of hyperbolic relative equilibria. The analysis is
analogous to the above. This time we put
my=Mcosh α, mz=M
ksinh α, (28)
so C(m) = −M2<0. Taking into account that mx= 0, then (22b) is satisfied provided
that (25), and hence also (26), hold. The condition for Min this case is
M2=µ1a2sinh3qU 0(q)
µsinh2αcosh q−sinh αsinh(q−α).(29)
12
It can be shown, using (26), that the right hand side of this expression is positive. There-
fore, upon the same considerations as above when counting the number of relative equi-
libria, we conclude
Proposition 3.2. There exists a unique family of hyperbolic relative equilibria that is
parametrised by q. This family has mx=p= 0, and myand mzdefined by (24)where α
and |M|are determined by (26)and (29).
This family of relative equilibria corresponds to the smooth bifurcation curve on the
semi-plane C < 0 on the energy – momentum diagram (see Fig. 4).
Non-existence of parabolic relative equilibria. Finally we show that there are
no solutions of (22b) having C(m) = 0. Substituting mx= 0 and my=±kmzinto the
first component of (22b) yields, after a simple calculation,
cosh 2q±sinh 2q+µ= 0,
which clearly has no solutions for qsince µ > 0.
This means that the bifurcation curves corresponding to elliptic and hyperbolic relative
equilibria meet each other at the punctured point when C= 0, see Fig. 4.
3.2 Stability analysis of the relative equilibria
Denote the level surface of Casimir function by
MM0={(m, q, p)|C(m) = M2
0}
and let the restriction of the Hamiltonian (20) be ˜
H=H|MM0. The relative equilibria
found above (Propositions 3.1, 3.2) are critical points of ˜
H.
According to Lyapunov’s Theorem, if a relative equilibrium is a local maxima or
minimum of ˜
H, it is nonlinearly stable. This happens in particular if the Hessian matrix
of ˜
Hat the equilibrium is positive or negative definite. On the other hand, if the Hessian
matrix of ˜
Hat an equilibrium has an odd number of negative eigenvalues, then the
linearised system has at least one eigenvalue with positive real part and the equilibrium
is unstable.
We now compute the signature of the Hessian matrix of ˜
Hat the relative equilibria
found in the previous section. This will give definite information on the nonlinear stability
of these solutions. The results in this section were obtained previously in [13] by working
on a symplectic slice in the unreduced system, since the reduced equations of motion were
not known at that time. The two approaches are in fact equivalent.
Analysis for the elliptic relative equilibria. Fix q0∈R+. According to Propo-
sition 3.1, there is a unique elliptic relative equilibrium (m, q, p)=(m0, q0,0) associated
to q0, with the vector m0given by
m0=0, M0sinh α0,M0
kcosh α0.
Here α0and M06= 0 are the values taken by αand Min (26) and (27) when one puts
q=q0.2Moreover, we have C(m0) = M2
0. The relative equilibrium (m0, q0,0) is a
critical point of
Hλ0(m, q, p) = H(m, q, p)−λ0
2C(m),
where the Lagrange multiplier is given by
λ0=cosh(q0−α0)
µ1a2sinh α0sinh q0
.
2there is an unessential choice in the sign of M0that we continue to ignore.
13
It can be shown that after an appropriate choice of coordinates the Hessian matrix N
of ˜
Hcoincides with the restriction of the quadratic form defined by the Hessian matrix
D2Hλ0(m0, q0,0) to the tangent space TMM0. If we choose the following basis vectors
for TMM0:
e1= (0,0,0,0,1),e2= (1,0,0,0,0),
e3= (0, k cosh α0,sinh α0,0,0),e4= (0,0,0,1,0),
then the resulting matrix Nhas the 2 ×2 block form
N=N(1)0
0N(2)
where
N(1)=1
µ1a2
k2(µ+ 1) −k
−kcosh α0cosh q0
sinh α0sinh q0
.
It is straightforward to check that N(1)is positive definite (recall that q0, α0>0).
Using (25) the entries of the matrix N(2)are written as
N(2)
11 =k2cosh q0
µ1a2sinh α0sinh q0cosh α0
,
N(2)
12 =N(2)
21 =kM0sinh(q0−2α0)−sinh(3q0−2α0)
2µ1a2sinh3q0
,
N(2)
22 =U00(q0) + M2
0cosh α0µcosh α0(2 cosh2q0+ 1) + cosh(α0−2q0) + 2 coshα0
µ1a2sinh4q0
.
A numerical study shows that for the gravitational potential (20), the trace of the
the matrix N(2)is positive for any value of q0. On the other hand, its determinant is
positive for for small q0and negative for large q0for any value of µ. We conclude3that
the signature of the Hessian at elliptic relative equilibria is (+,+,+,+) for small q0and
(+,+,+,−) for larger values of q0. The former equilibria are Lyapunov stable while the
latter are unstable. The system undergoes a saddle-node bifurcation. This corresponds
to the cusp in Figure 4.
As shown in Figure 4, for a given µthe change of stability of the elliptic relative equi-
libria occurs at a maximum value of M2given by (29). An implicit plot of dM 2
dq (q, µ)=0
leads to the diagram 3for the stability of elliptic relative equilibria.
Analysis of the hyperbolic relative equilibria. Proceeding in a completely anal-
ogous way as for the elliptic relative equilibria, one obtains the corresponding matrix N
that is block diagonal with the blocks N(1)and N(2). This time N(1)is given by
N(1)=1
µ1a2
k2(µ+ 1) −k
−kcosh q0sinh α0
sinh q0cosh α0
,
that may be shown to be positive definite using (26).
3For an analytical proof see [13].
14
Figure 3: Stability region for elliptic relative equilibria as functions of qand the mass ratio
µ=µ1/µ2. For a given mass ratio µ, the equilibria are stable for small qand unstable for
large q.
On the other hand, using (26), the entries of N(2)can be written as
N(2)
11 =k2cosh q0
µ1a2sinh q0sinh α0cosh α0
,
N(2)
12 =N(2)
21 =kM0cosh q0−cosh(q0−4α0)−cosh 3q0+ cosh(3q0−4α0)
4µ1a2sinh3q0sinh 2α0
,
N(2)
22 =U00(q0) + M2
0sinh α0µsinh α0(2 cosh2q0+ 1) + sinh α0−2 cosh q0sinh(q0−α0)
µ1a2sinh4q0
.
A numerical study4shows that for the gravitational potential the determinant of N(2)
is negative for any value of q0. Therefore, its eigenvalues have opposite sign. In conclusion,
the signature of the Hessian matrix along the branch of hyperbolic relative equilibria is
(+,+,+,−) and hence, they are all unstable.
Energy – momentum diagram. In the case of the gravitational potential (21),
our analysis is summarised in the bifurcation diagram given in Figure 4. The shaded
region corresponds to the image of the energy momentum mapping (C, H ) from the phase
space of the system into R2. The elliptic relative equilibria are indicated in blue. Those
having signature (+,+,+,+) are stable and correspond to small values of q(the particles
are close). Eventually, when the distance between the particles is sufficiently large, the
momentum Cof the elliptic relative equilibria achieves a maximum, and there is a change
in the stability. This corresponds to the cusp in the figure. The elliptic RE with signature
(+,+,+,−) correspond to large values of qand are all unstable. They approach the
axis C= 0 as q→ ∞. The hyperbolic RE are illustrated in red. They have signature
(+,+,+,−) and are unstable. They also approach the axis C= 0 as q→ ∞. Notice that
the families seem to meet when C= 0 but that there is no relative equilibrium at this
point, since, as we have shown, there do not exist parabolic relative equilibria.
The value of Hwhere the two families seem to meet, and that creates the boundary
of the shaded area for large C > 0 is equal to the limit of the potential Uas q→ ∞.
4An analytic proof is given in [13].
15
Figure 4: Energy-Momentum bifurcation diagram of relative equilibria for the gravitational
potential with G= 2, a = 1, k = 1, µ1= 0.5, µ2= 1. The shaded area on the C-Hplane
shows all possible values of (C, H). We also indicate the signature of the Hessian matrix of
the Hamiltonian along each branch of relative equilibria. Notice the change in signature at
the cusp of the elliptic relative equilibria where a saddle-node bifurcation takes place.
3.3 Topology of the Energy-Momentum level surfaces
Denote by Zthe reduced phase space. Z∼
=R4×R+with global coordinates m, p, q. The
map (C, H ) : Z → R2is not proper. It has the following properties:
Proposition 3.3. Let ζn= (mn, pn, qn)be a sequence on Zand suppose that (C, H)(ζn)
converges to (c0, h0)∈R2. Then
(i) If qn→ ∞ then h0>limq→∞ U(q).
(ii) If qn→0then c060.
(iii) If ε6qn61
εfor some ε > 0, then mnand pnare bounded.
Proof. (i) Since the kinetic energy is positive, we have H(ζn)>U(qn), and the result
follows by letting n→ ∞.
(ii) Suppose −m2
x−m2
y+k2m2
z=c0>0. Then, for nlarge enough, k2(mz)2
n>c0
2and
we have
H(ζn)>c0µ
4µ1a2sinh2qn
−Gµ1µ2coth qn
that grows without bound as qn→0. This contradicts our hypothesis that H(ζn)
converges to h0.
(iii) For ε6q61
εit is possible to bound Hfrom below by a constant, positive definite
quadratic form on mnand pnwith constant coefficients, plus a constant value (the
minimum of Ufor ε6q61
ε). Hence, the only way in which Hcan remain bounded
if qis bounded is if mnand pnare also bounded.
16
One can show that the fibre over the point (c0, h0) is compact only for c0>0 and
h0<limq→∞ U(q). In this case it topologically consists of the disjoint union of two
3-spheres. We can in fact describe the topology of the fibre for all other values (c0, h0)
having c0>0. For the small region inside the cusp it is these two 3-spheres with two
unbounded 3-dimensional balls, and for (c0, h0) the upper region of the C-Hplane it is
just the two unbounded 3-dimensional balls.
We were unable to determine the topology of the unbounded fibres having c0<0.
The transition from c0>0 to c0<0 is surprisingly complicated.
4 Relative equilibria for the 2-body problem on the
sphere
Denoting µ=µ1
µ2
,q1=qand p1=p, the reduced Hamiltonian (13) takes the form
H=1
2a2µ1(m,A(q)m)+2mxp+ (1 + µ)p2+U(q),(30)
where q∈(0, π) is the angle between the radius vector of the two particles and
A(q) =
1 0 0
0 1 cos q
sin q
0cos q
sin q
µ+ cos2q
sin2q
.
Since the symmetry group SO(3) is compact and has rank 1, all of the relative equi-
libria of the problem for (m,m)6= 0 are periodic solutions. The 2 masses simultaneously
rotate about a fixed axis of rotation at a steady angular speed ω.
Relative equilibria are equilibrium points of the reduced system and are characterised
as critical points of Hon a level set (m,m) = M2
0. Therefore, at such solution the
following equations must be satisfied:
∂H
∂p = 0,(31a)
∂H
∂m×m=0,(31b)
∂H
∂q = 0.(31c)
The condition (31a) yields
p=−mx
1 + µ.(32)
Substituting (32) into (31b) one obtains two possibilities:
(i) my=mz= 0. In this case (31c) implies dU
dq = 0. If the potential is attractive there
is no solution.
(ii) mx= 0. We focus on this case in what follows.
Introduce the angle θby setting
mz=M0cos θ, my=M0sin θ. (33)
In order to interpret θ, consider the reconstructed motion along the relative equilibrium.
As in subsection 1.3, assume that the fixed axes OXY Z are chosen in such a way that
17
M||OZ. Then, in view of (9) and given that mx= 0, the angle θcoincides with the Euler
angle θof the matrix Qif we take the Euler angle ϕ= 05. As it follows from (10), both of
these angles remain constant along the relative equilibrium. On the other hand we have
˙
ψ=ωwhere the constant angular velocity ωis computed from (14) to be
ω=M0cos(q−θ)
a2µ1sin qsin θ.(34)
Now recall from (12) that the positions of the masses on the moving frame are r1= (0,0, a)
and r2= (0, a sin q, a cos q). Assuming ψ(0) = 0, the positions Rα=Qrαof the particles
on the fixed axes are
R1(t) = a
sin ωt sin θ
−cos ωt sin θ
cos θ
,R2(t) = a
−sin ωt sin(q−θ)
cos ωt sin(q−θ)
cos(q−θ)
.
Therefore, the two particles rotate about the OZ axis with angular velocity ωas illustrated
in Figure 5below, and θis the angle between the particle µ1and the fixed axis of rotation.
Hence, without loss of generality we may assume that 0 6θ6π
2.
Figure 5: 0 6θ6π/2 is the angle that the first particle makes with the axis of rotation.
Substitution of (33) into (31b) yields the necessary condition
µsin(2θ)−sin(2(q−θ)) = 0,(35)
that is equivalent to the “law of lever” requirement given in [7]. We will analyse separately
the case of equal and different masses.
4.1 The case of different masses
Assume that the masses are different and, without loss of generality, that µ1< µ2(i.e.
0<µ<1). With our assumption that 0 6θ6π
2equation (35) implicitly defines θas a
smooth function of qon the open intervals (0,π
2) and (π
2, π). Figure 6shows a graph of
θ=θ(q).
The interpretation is the following: suppose that we are given a relative configuration
specified by an angle q∈(0, π), q6=π
2and we ask ourselves if there is a relative equilibrium
at this relative configuration. Below we show that such relative equilibrium does exist, it
is unique (up to the sense of rotation), and that equation (35) specifies the unique angle
θ∈(0,π
2) between the particle µ1and the corresponding axis of rotation. The relative
configurations having q=π
2require a more careful analysis. As we shall see, for the
gravitational potential, they do not admit relative equilibria.
5The other possibility, namely that ϕ=π, simply changes the sign of θ.
18
Figure 6: The angle θas a function of qfor µ= 0.7.
Substitution of (33) and (32) into (31c) leads to
M2
0=µ1µ2a2sin3q U0(q)
cos θ(µ1cos qcos θ+µ2cos(q−θ)).(36)
For an attractive potential we have U0(q)>0 so, for any q∈(0, π), q6=π
2, equation (36)
determines a unique value of |M0|that specifies the speed of rotation ωby (34) (in all
expressions θ=θ(q) as explained above).
Once again, the choice of sign for M0corresponds to two solutions related by reversing
the direction of time. We shall not distinguish these two. Therefore, we have shown the
following.
Proposition 4.1. Suppose that µ16=µ2. For any relative configuration q∈(0, π),q6=π
2,
there exists a unique relative equilibrium; it has axis of rotation specified by the unique
angle θ∈(0, π/2) satisfying (35), and a unique speed of rotation ωspecified by (34)where
M2
0is given by (36). Along these solutions mx= 0 and p= 0.
Relative equilibria with relative configuration q∈(0, π/2) will be called acute rela-
tive equilibria. For these we may write q=θ+1
2arcsin(µ(sin 2θ)) and the inequality
θ > q
2holds. Therefore, they are characterised by the condition that the heavier mass
is closer to the axis of rotation. They correspond to the smooth bifurcation curve in the
Energy-Momentum diagram 7(a).
Relative equilibria having relative configuration with q∈(π/2, π) will be called obtuse
relative equilibria. For these we have q=θ+π
2−1
2arcsin(µ(sin 2θ)) and they satisfy θ <
q
2. For these, it is the the lighter mass that is closer to the axis of rotation. These relative
equilibria correspond to the bifurcation curve with the cusp in the Energy-Momentum
diagram 7(a).
For the gravitational potential U(q) = −Gµ1µ2cot(q), for a constant G > 0, and
equation (36) becomes
M2
0=Gµ2
1µ2
2a2sin q
cos θ(µ1cos θcos q+µ2cos(θ−q)).(37)
In view of (35) the angle θapproaches π/2 as q→π/2 from the left and it approaches
0 as q→π/2 from the right. In both cases, according to (37), M2
0→ ∞. Therefore,
in the gravitational case there do not exist relative equilibria having q=π
2. It follows
19
that for the gravitational potential, acute and obtuse relative equilibria are two separate
branches. Acute relative equilibria resemble their euclidean counterpart, whereas obtuse
relative equilibria do not exist in the zero curvature case.
4.2 The case of equal masses
If µ1=µ2then µ= 1 and equation (35) is only satisfied in the following two different
situations that lead to two families of relative equilibiria.
(i) The first family is labeled6by θ∈(0, π/2) and has q=π/2. We will refer to the
members of these family as right-angle relative equilibria.
(ii) For the second family q= 2θ. These will be called isosceles relative equilibria.
Depending on the value of q∈(0, π), we speak of acute, obtuse or right-angle
isosceles relative equilibria.
For both families, the corresponding value of M2
0is determined by substituting the
values of qand θin (37). Note that both families intersect at the right angle-isosceles
relative equilibrium as illustrated in the Energy-Momentum diagram in Figure 7(b). These
considerations, together with the analysis of the previous section allow us to conclude the
following.
Proposition 4.2. Suppose that µ1=µ2.
(i) Consider the relative configuration q=π
2. Given any θ∈(0, π /2) there exists a
unique relative equilibrium in which the first mass makes an angle θwith the axis of
rotation.
(ii) For any relative configuration q∈(0, π),q6=π
2, there exists a unique relative
equilibrium. The axis of rotation in such relative equilibrium makes an angle θ=q/2
with any of the particles.
In both cases, the unique speed of rotation ωis specified by (34)where M2
0is given by
(36). Along both of these families mx= 0 and p= 0.
As usual, in the statement of the proposition, we do not distinguish between relative
equilibria having opposite directions of rotation.
4.3 The signature of the Hessian of the reduced Hamiltonian
along the relative equilibria
Let
Hλ(m, q, p) = H(m, q, p)−λ
2(m,m).
A relative equilibrium (m0, q0,0) = (0, M0sin θ0, M0cos θ0, q0,0) found in the previous
section is a critical point of Hλif the Lagrange multiplier
λ=λ0=cos(q0−θ0)
µ1a2sin θ0sin q0
.
To compute the Hessian matrix of the restriction of Hto the level set (m,m) = M2
0at the
relative equilibrium, we proceed as in section 3.2. Namely, we consider the restriction of
the quadratic form defined by the Hessian matrix D2Hλ0(m0, q0,0) to the tangent space
of the surface (m,m) = M2
0at the critical point. In terms of the basis
e1= (0,0,0,1,0),e2= (1,0,0,0,0),
e3= (0,−cos θ0,sin θ0,0,0),e4= (0,0,0,0,1),
6It could be labeled by θ∈(0, π/4] by introducing a Z2particle-relabeling symmetry.
20
the resulting matrix Nhas the 2 ×2 block form
N=N(1)0
0N(2)
where
N(1)=1
µ1a2
(µ+ 1) 1
1−cos θ0cos q0
sin θ0sin q0
.
Assume for the moment that the masses are different. It is straightforward to check that
det(N(1))<0 for acute relative equilibria since for these 0 < θ0, q0<π
2. The same
inequality holds for obtuse relative equilibria as can be shown using (35) and the estimate
q−π
2< θ < q
2that follows from (35) for q∈(π/2, π), and θ∈(0,π
2). Therefore, the
signature of N(1)is (+,−) both for the obtuse and the acute relative equilibria.
The entries of the matrix N(2)are
N(2)
11 =−2 cos q0
µ2a2sin(2(q0−θ)) sin q,
N(2)
12 =N(2)
21 =−M0(µsin 2θ0cos q0+ sin(2θ0−q0))
µ1a2sin3q0
,
N(2)
22 =U00(q0) + M2
0cos θ0(µ+ 1) cos θ0(2 cos2q0+ 1) + sin θsin 2q0
µ1a2sin4q0
.
where we have used (35) to simplify the expression of N(2)
11 .
A numerical study of the signature of N(2)for the gravitational potential is easily
performed by writing q0in terms of θ0with the corresponding expressions for acute or
obtuse relative equilibria given above. Such study shows that the determinant of N(2)
is always negative for acute relative equilibria. On the other hand, for obtuse relative
equilibria, the determinant of N(2)passes from being negative for small θto being positive
for larger values of θ. The trace of N(2)in this case is positive, so we conclude that the
signature of N(2)for obtuse RE changes from (+,−) to (+,+) as θincreases.
Summarising, our numerical studies show that the signature of the restricted Hessian
for all acute relative equilibria is (+,+,−,−). For the obtuse relative equilibria, given that
θis an increasing function of q, we conclude the signature is (+,+,−,−) for π
2< q < q∗
and (+,+,+,−) for q∗< q < π for a certain obtuse critical angle q∗that only depends
on the ratio between the masses.7The change of signature corresponds to the cusp in the
bifurcation curve of the Energy-Momentum diagram 7.
The analysis is of the signature of the matrices N(1)and N(2)simplifies significantly if
the masses are equal. For the gravitational potential, we have for the right-angle relative
equilibria
det(N(1)) = −1
a4µ2
1
<0,det(N(2)) = −2Gµ1cos22θ
a2sin 2θ.
Note that det(N(2))<0 for all 0 < θ < π/2 except for θ=π
4where it vanishes. On the
other hand, for the isosceles relative equilibria one obtains
det(N(1)) = −cot2(q/2)
a4µ2
1
<0,
det(N(2)) = −Gµ1(3 −2 cos2(q/2))(2 cos2(q/2) −1)
16a2cos5(q/2) sin5(q/2) ,
trace(N(2)) = Gµ3
1a2−4 sin(q/2) cos2(q/2)(2 cos2(q/2) −1)
8a2µ1cos5(q/2) sin3(q/2) .
7An analytic proof of of this was recently obtained in [18].
21
It is clear that det(N(2)) is negative for q∈(0, π/2) and positive for q∈(π/2, π). For the
latter values of q, we have trace(N(2))>0.
Therefore, the signature of the restricted Hessian for acute isosceles relative equilibria
is (+,+,−,−), while obtuse ones have (+,+,+,−). The change occurs at the right-angle
isosceles relative equilibrium where the family of isosceles relative equilibria intersects
the family of right-angle relative equilibria, see Figure 7(b). Away from this intersection
point, the signature of right-angle relative equilibria is (+,+,−,−).
Stability. All relative equilibria of the problem for which the signature of the re-
stricted Hessian is (+,+,+,−) are unstable. The question remains to determine the
stability of those having signature (+,+,−,−). All of these are elliptic equilibria (see [18]
and section 4.4 ahead) so they are candidates to be stable solutions.
In section 4.4 below we will perform an analysis of stability for acute RE that involves
Birkhoff normal forms and the use of KAM techniques. Our treatment suggests that
generic acute RE are nonlinearly stable.
Energy-Momentum Diagram. Below we reproduce the energy-momentum diagram
of the system given in [7] for the gravitational potential. We complement the diagram
by illustrating the image of the energy-momentum map (M2, H) and the signature of the
restriction of the Hessian to the constant momentum surfaces along each family of relative
equilibria. In view of the discussion above, such diagram must be done separately for the
cases of equal and different masses.
(a) The case of different masses, µ1=
0.5, µ2= 1. Notice the change in sig-
nature at the cusp of the obtuse relative
equilibria
(b) The case of equal masses, µ1=
µ2= 1/√2. There is a change of sig-
nature passing from acute to obtuse
isosceles RE when the two families of
RE intersect.
Figure 7: Energy-Momentum bifurcation diagram of relative equilibria for the gravitational
potential with G= 2, a = 1, µ1= 0.5, µ2= 1. The shaded area on the M2-Hplane show all
possible values of (M2, H).
Remark 4.3. Considering that the gravitational potential Usatisfies limq→0U(q) = −∞,
it is easy to construct a sequence qn, pnwith qn→0, and pn→ ∞, such that Hevaluated
at (m, q, p) = (0, M0,0, qn, pn) converges to any arbitrary value h0∈R. This shows
that the energy-momentum map (M2, H) is not proper and that all of its fibres are non-
compact.
22
4.4 Stability of acute relative equilibria in the case of different
masses
The analysis presented in the previous section shows that one cannot prove stability of
RE for the 2-body problem on S2under the gravitational potential relying on purely
topological methods. However, the restriction of the reduced equations of motion to the
surface of constant momentum (m,m) = M2>0, that we denote by MM2, is a two-
degree of freedom Hamiltonian system. Even if topological methods are not applicable,
sufficient conditions for nonlinear stability of equilibria for these kind of systems may be
obtained by computing Birkhoff normal forms and applying KAM techniques as we now
recall.
Consider a RE of the problem that projects to an (isolated) equilibrium point on
MM2. Our investigation of its stability will proceed by checking that the following three
conditions are satisfied:
1◦. It is an elliptic equilibrium point of the restriction of the reduced system to MM2.
Namely, the eigenvalues of the linearized system are purely imaginary
λ1=iΩ1, λ2=−iΩ1, λ3=iΩ2, λ4=−iΩ2,0<Ω1<Ω2.
It is well known that this is a necessary necessary condition for stability (in the absence
of zero eigenvalues). Next we will check that
2◦. there are no second or third-order resonances:
Ω26= 2Ω1,Ω26= 3Ω1.
Under this condition one may put the Hamiltonian (restricted to MM2) in Birkhoff normal
form
H=1
2
2
X
j=1
αjIj+1
4
2
X
j,k=1
βjk IjIk+O5, Ij=x2
j+y2
j,|αi|= Ωi.(38)
Here, xjand yjare suitable canonical coordinates on a neighbourhood of the equilibrium
on MM2(i.e., {xj, yk}=δjk ) with the equilibrium located at xj=yj= 0, βj k are
constants, and O5denotes a power series containing terms of order no less that 5 in
xj, yj.
If conditions 1◦and 2◦are satisfied, a sufficient condition for nonlinear stability (un-
der perturbations within MM2) may be given in terms of the nonlinear terms in (38).
Specifically, one requires that
3◦.the Arnold determinant is different from zero
D:= det
β11 β12 α1
β12 β22 α2
α1α20
= 2β12α1α2−β11 α2
2−β22α2
16= 0.
This nonlinear condition allows one to apply the KAM theorem in such a way that the
invariant tori act as boundaries for the flow on each constant energy surface, leading to
Lyapunov stability of the equilibrium (see e.g. [17], §35 in [19], or Section 13 in [15] for
proofs and details).
Remark 4.4. If the Arnold determinant D= 0, one may still obtain sufficient conditions
for stability by considering higher order terms in the normal form expansion (38) (see
e.g. [15]). On the other hand, the presence of second or third-order resonances may lead
to instability. We shall not consider any of these possibilities here.
Remark 4.5. We emphasise that the above analysis ensures nonlinear stability of RE
only with respect to perturbations on the initial conditions that lie on the momentum
surface MM2.
23
Below we analyse these conditions for the acute RE presented in Section 4.1. We give
numerical evidence that suggests that they are generically nonlinearly stable.
In our analysis we assume that the constants κ1:= Gµ1µ2and κ2:= a2µ1equal
one. In this way, the gravitational potential U(q) = −cot(q), and the Hamiltonian (30)
depends on the parameters of the problem only through the mass ratio µthat we will
continue to assume to be 0 <µ<1.8Therefore, in view of Proposition 4.1, the stability
of the RE of the problem depends on two essential parameters. An internal parameter
that labels the RE of the problem, and the external parameter µ.
Linear analysis
Introduce cylindrical coordinates (z, θ ) on MM2
mx=z, my=pM2−z2sin θ, mz=pM2−z2cos θ.
Then (q, p, θ, z ) are Darboux coordinates and the restriction of the reduced equations of
motion to MM2takes the canonical form
˙q=∂H
∂p ,˙p=−∂H
∂q ,˙
θ=∂H
∂z ,˙z=−∂H
∂θ ,
where, in view of (30),
H=1
2"(1 + µ)p2+ 2pz −cos 2q+ cos 2(q−θ) + µ(1 + cos 2θ)
2 sin2qz2
−1
sin2qsin 2q−M2
21 + cos 2(q−θ) + µ(1 + cos2θ)#.
Of course, in the above equations, M2should be treated as a constant.
The RE of the problem are the equilibria of the above equations. As it was shown in
Section 4.1, these occur at points where p= 0, z= 0, θand qare related by (35). In
addition, in view of (37), qand θshould be such that
M2=sin q
cos θcos(q−θ) + µcos θcos q.(39)
Consider the RE described above, and a small deviation from it within MM2given by
the vector ∆= (∆z, ∆p, ∆θ, ∆q). The linearized system has the form
˙
∆=A∆,A=0A(1)
A(2) 0,
where A(1) and A(2) are symmetric 2 ×2 matrices. The entries of A(1) may be written as
A(1)
11 =M2(cos(2(q−θ)) + µcos 2θ)
sin2q,A(1)
22 =−M2(1 + µ) cos2θ
sin4q,
A(1)
12 =A(1)
21 =M2(sin qcos 2θ−(1 + µ) cos qsin 2θ)
sin3q,
(40)
8The assumption that κ1and κ2equal 1 is done without loss of generality since one may eliminate these
quantities from the equations of motion (8) by rescaling time t→κ2t
√κ1, and the momenta p→√κ1p, m→
√κ1m.
24
where we have used (39) to simplify A(1)
22 . On the other hand
A(2) =
−cos θcos θcos 2q+ sin θsin 2q+µcos θ)
sin2q1
1 1 + µ
.(41)
The eigenvalues of of the matrix Aare the roots of its characteristic polynomial, which
due to the Hamiltonian nature of the problem turns out to be biquadratic:
P(λ) = λ4+aλ2+b.
Proposition 4.6. All acute RE are elliptic.
Proof. The ellipticity condition 1◦may be written in terms of the coefficients aand bof
the characteristic polynomial Pas
a > 0, b > 0, R1:= 1
4a2−b > 0.(42)
Using the expressions given above for the entries of Aand (35) and (39) we can simplify
a=M2(1 + cos(2(q−θ)))
sin2qsin2θ, b =a2
41−4 sin2θsin2(q−θ),
R1=a2sin2θsin2(q−θ).
(43)
Recall from section 4.1 that acute RE correspond to the solutions of (35) that satisfy
q=θ+1
2arcsin(µ(sin 2θ)),(44)
with θ∈(0, π/2). Using this, it is immediate to see that the inequality a > 0, and hence
also R1>0, hold. On the other hand, the inequality b > 0 in (42) is equivalent to
f:= 1 −4 sin2θsin2(q−θ)>0.(45)
To show that this inequality holds, we write, using (44),
f= 1 −2 sin2θ1−q1−µ2sin22θ.
Since µ2sin22θ < sin22θfor all µ∈(0,1), we find that
f > 1−2 sin2θ(1 − | cos 2θ|) = (1−4 sin4θ0< θ 6π/4,
cos22θ π/46θ < π/2.
The above function is everywhere greater than zero except at the point θ=π/4, but, as
can be verified,
fθ=π/4=p1−µ2>0.
Remark 4.7. An analogous analysis may be performed to show that the obtuse RE with
signature (+,+,−,−) are elliptic. For these one has
q=θ+π
2−1
2arcsin(µ(sin 2θ)),(46)
and, using (43), one may easily verify that a > 0 (and hence also R1>0) for all θ∈
(0, π/2), µ ∈(0,1). On the other hand, the corresponding expression for fis f= 1 −
25
2 sin2θ1 + p1−µ2sin22θ. This is a strictly decreasing function of θon the interval
(0, π/2), that is positive for 0 < θ < θ∗and negative for θ∗< θ < π/2. Here θ∗is the
unique root of the equation
cos 2θ= 2 sin2θq1−µ2sin22θ
on the interval (0, π/2). This value of θ∗corresponds to the cusp in the bifurcation
diagram Fig. 7(a), where the signature of the obtuse RE changes.
Figure 8below illustrates the plane a-bof coefficients of the characteristic polynomial
of A. The curves σ1and σ2respectively correspond to the values of (a, b) attained at
the acute and obtuse RE of the problem for the fixed value of µ= 0.95. These RE are
conveniently parametrised by θ∈(0, π/2) by respectively using (44) and (46). The figure
also illustrates the parabolae corresponding to the zero loci of R1, and of the second and
third order resonance polynomials R2and R3introduced below.
Figure 8: Curves σ1,σ2, corresponding to acute and obtuse RE (respectively given by (44)
and (46)) on the coefficient plane (a, b) for µ= 0.95.
Resonances
Condition 2◦that requires that there are no second or third-order resonances is written
in terms of the coefficients aand bof the characteristic polynomial of Aas
R2:= 4
25a2−b6= 0, R3:= 9
100a2−b6= 0.(47)
An analytic investigation of these conditions involves very heavy calculations so we present
numerical results. We restrict our attention to the acute RE that may be parametrised
by θusing (44), and we present our stability results in terms of the parameters (µ, θ)∈
(0,1) ×(0, π/2).
One can express R2and R3as functions of (µ, θ) by using (43), (39) and the relation
(44) for acute RE. The zero loci of R2and R3on the θ-µ-plane are the two curves
illustrated in Fig. 9.
26
Analysis of the Arnold determinant
As for the resonance condition, we only present numerical results for our investigation of
condition 3◦for the acute RE. By using (39) and (44), we express D=D(µ, θ).
The zero locus of Don the plane θ-µconsists of the two curves illustrated in Figure 9.
Figure 9: Curves on the plane µ-θ0plane corresponding to RE with second and third order
resonances (respectively R2= 0 and R3= 0) and where the Arnold determinant vanishes
(D= 0).
Nonlinear stability of acute RE
According to the discussion above, the acute RE that correspond to parameter values
(µ, θ) lying outside of the curves in Fig. 9are nonlinearly stable. Therefore, we have
provided numerical evidence to show that generic9acute RE are stable in the sense of
Lyapunov.
The stability analysis for the RE corresponding to the exceptional parameter values
(µ, θ) that lead to resonances R2= 0 and R3= 0, or the vanishing of the Arnold deter-
minant D= 0, has to be done separately.
Open problems
To conclude, we point out a number of open problems concerning the two-body problems
in spaces of constant curvature:
– In the case of different masses in S2, investigate the nonlinear stability of acute RE
for which there are resonances (R2= 0, R3= 0) and/or the Arnold determinant
vanishes (D= 0).
– In the case of different masses in S2, investigate the nonlinear stability of obtuse RE
for which the signature of the reduced Hamiltonian is (+,+,−,−).
– In the case of equal masses in S2, investigate the nonlinear stability of acute-isosceles
RE and right-angle RE.
– Classify and investigate the stability of all RE for the spatial two-body problem on
S3and L3.
9for an open dense set of parameter values.
27
Acknowledgements
We are grateful to Miguel Rodr´ıguez-Olmos for discussing his preliminary results of [18]
with us. The authors express their gratitude to B. S. Bardin and I. A. Bizyaev for fruitful
discussions and useful comments.
The research contribution of LGN and JM was made possible by a Newton Advanced
Fellowship from the Royal Society, ref: NA140017.
The work of AVB and ISM is supported by the Russian Foundation for Basic Re-
search (project No. 17-01-00846-a). The research of AVB was also carried out within the
framework of the state assignment of the Ministry of Education and Science of Russia.
References
[1] Borisov A.V., Mamaev I.S., Rigid body dynamics. Hamiltonian methods, integrabil-
ity, chaos. Moscow–Izhevsk: Institute of Computer Science, 2005, 576 p. (in Russian).
[2] Borisov A.V., Mamaev I.S., Reduction in the two-body problem on the Lobatchevsky
plane. Russian Journal of Nonlinear Dynamics, 2006, vol.2, no.3, pp.279–285 (in
Russian).
[3] Markeev A.P., Libration points in celestial mechanics and cosmodynamics. M.:
Nauka, 1978, 312 p. (in Russian).
[4] Borisov A.V., Mamaev I.S., Rigid Body Dynamics in Noneuclidean Spaces. Russian
Journal of Mathematical Physics, 2016, vol.23, no.4, pp.431–453.
[5] Borisov A.V., Mamaev I.S., The Restricted Two-Body Problem in Constant Curva-
ture Spaces. Celestial Mech. Dynam. Astronom., 2006, vol.96, no.1, pp.1–17.
[6] Borisov A.V., Mamaev I.S., Bizyaev I.A., The Spatial Problem of 2 Bodies on a
Sphere. Reduction and Stochasticity. Regular and Chaotic Dynamics, 2016, vol.21,
no.5, pp.556–580.
[7] Borisov A.V., Mamaev I.S., Kilin A.A., Two-Body Problem on a Sphere: Reduction,
Stochas-ticity, Periodic Orbits. Regul. Chaotic Dyn., 2004, vol.9, no.3, pp.265–279.
[8] Bolsinov A.V., Borisov A.V., Mamaev I.S., Topology and stability of integrable sys-
tems. Russian Mathematical Surveys, 2010, vol.65, no.2, pp.259–318.
[9] Bolsinov A.V., Borisov A.V., Mamaev I.S., The Bifurcation Analysis and the Conley
Index in Mechanics. Regular and Chaotic Dynamics, 2012, vol.17, no.5, pp.457–478.
[10] Chernoivan V.A., Mamaev I.S., The Restricted Two-Body Problem and the Kepler
Problem in the Constant Curvature Spaces. Regular and Chaotic Dynamics, 1999,
vol.4, no.2, pp.112–124.
[11] Diacu F., Relative Equilibria of the Curved N-Body Problem. Paris: Atlantis, 2012.
[12] Diacu F., Prez-Chavela E., Reyes J.G., An intrinsic approach in the curved n-body
problem. The negative case. J. Differential Equations, 2012, vol.252, pp.4529–4562.
[13] Garc´ıa-Naranjo L.C., Marrero J.C., P´erez-Chavela E. and Rodr´ıguez-Olmos M., Clas-
sification and stability of relative equilibria for the two-body problem in the hyper-
bolic space of dimension 2. J. of Differential Equations, 2016, vol.260, pp.6375–6404.
[14] Kilin A.A., Libration Points in Spaces S2and L2. Regular and Chaotic Dynamics,
1999, vol.4, no.1, pp.91–103.
[15] Meyer K.R., Hall G.R, and Offin, D., Introduction to Hamiltonian dynamical sys-
tems and the N-body problem. Second edition. Applied Mathematical Sciences, 90.
Springer, New York, 2009.
28
[16] Montanelli H., Computing Hyperbolik Choreographies. Regular and Chaotic Dynam-
ics, 2016, vol.21, no.5, pp.523–531.
[17] Moser J., Lectures on Hamiltonian systems. American Mathematical Soc., 1968,
no.81.
[18] Rodr´ıguez-Olmos M., Relative equilibria for the two-body problem on S2. In prepa-
ration.
[19] Siegel C.L. and Moser J.K., Lectures on celestial mechanics. Translated from the
German by C. I. Kalme. Reprint of the 1971 translation. Classics in Mathematics.
Springer-Verlag, Berlin, 1995.
[20] Shchepetilov A.V., Two-Body Problem on Spaces of Constant Curvature: 1.Depen-
dence of the Hamiltonian on the Symmetry Group and the Reduction of the Classical
System. Theoret. and Math. Phys., 2000, vol.124, no.2, pp.1068–1081.
[21] Shchepetilov A.V., Calculus and Mechanics on Two-Point Homogenous Riemannian
Spaces. Lect. Notes Phys., vol.707, Berlin:Springer, 2006.
A.V. Borisov
Udmurt State University,
ul. Universitetskaya 1, Izhevsk, 426034 Russia
and
A.A.Blagonravov Mechanical Engineering Research Institute of RAS
ul. Bardina 4, Moscow, 117334 Russia
borisov@rcd.ru
L.C. Garc´ıa-Naranjo
Departamento de Matem´aticas y Mec´anica IIMAS-UNAM
Apdo. Postal: 20-726 Mexico City, 01000, Mexico
luis@mym.iimas.unam.mx
I.S. Mamaev
Institute of Mathematics and Mechanics of the Ural Branch of RAS
ul. S.Kovalevskoi 16, Ekaterinburg, 620990 Russia
and
Izhevsk State Technical University
Studencheskaya 7, Izhevsk, 426069 Russia
mamaev@rcd.ru
J. Montaldi
School of Mathematics, University of Manchester
Manchester M13 9PL, UK
j.montaldi@manchester.ac.uk
29
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