Collision trajectories and regularisation of two-body problem onS2 Alessandro Arsie Nataliya A. Balabanova

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Collision trajectories and regularisation of two-body problem

on S2

Alessandro Arsie & Nataliya A. Balabanova

August 2022

Abstract

In this paper, we investigate collision orbits of two identical bodies placed on the surface

of a two-dimensional sphere and interacting via an attracting potential of the form V(q)=

−cot(q), where qis the angle formed by the position vectors of the two bodies. We describe

the ω-limit set of the variables in the symplectically reduced system corresponding to initial

data that lead to collisions. Lastly, we regularise the system and investigate its behaviour on

near collision orbits. This involves the study of completely degenerate equilibria and the use

of high-dimensional non-homogenous blow-ups.

Keywords: Hamiltonian systems, Dynamical systems, Collisions, Regularisation, Weighted blow-

up, Completely Degenerate Equilibria.

Contents

1 Introduction 1

2 Setup 4

3 Singular trajectories 6

4 Regularisation 18

5 Conclusion 27

A Appendix 27

1 Introduction

In the standard formulation of the N-body problem on any surface Mthe phase space is taken

to be T∗(MN\∆), where ∆is the set of all N-tuples of points with two or more coinciding entries.

Putting it in a more straightforward way, the standard formulation of the N-body problem excludes

any collision.

This non-compact phase space arrangement immediately begs the question of how the bodies

behave when on collision orbits or in their immediate vicinity. A possible approach to these ques-

tions include ﬁnding the ω−limit set of the variables of the system as it tends to a collision and the

regularisation of the Hamiltonian vector ﬁeld.

Naturally, the ﬁrst works of this kind discuss the Kepler problem in the plane: see [13] for the

general overview and e.g. [19,17] for details.

arXiv:2210.13644v3 [math.DS] 8 Sep 2023

The Kepler problem on the plane is equivalent to the two-body problem. However, when the

curvature of the surface on which the bodies are moving is nonzero, things are rather different.

The Kepler problem becomes a separate (integrable) case, where one body, usually of mass 1 after

normalisation is ﬁxed, and the other is moving freely. In contrast, the two-body problem is not in-

tegrable and involves two freely-moving bodies of arbitrary masses. Collision orbits, Hill’s regions

and near collision dynamics in the former were described in [1]. Additional noteworthy result is

the regularisation of the vector ﬁeld near a collision point.

In [7], Borisov and Mamaev investigate the collisions in a speciﬁc case of the two-body problem

on a sphere, when the two bodies are restricted to move along a great circle. They demonstrate that

the collision time is ﬁnite, and that suitable regularisation yields elastic collisions.

A signiﬁcant percentage of the works dedicated to N-body problems on surfaces of constant

curvature can be roughly divided into two lines of investigation: the more laborious approach of

using the stereographic projection (e.g. [8,9]) and that of Marsden-Weinstein reduction ([2,21,7]

and many more).

In this paper, we follow the latter approach: attempting to perform stereographic projection

and describe collision orbits in eight dimensions proves to be a nearly insurmountable computa-

tional task.

The most important results on which we build for our work are contained in [5]: namely,

the method of reduction of our eight-dimensional system to a ﬁve-dimensional one through the

SO(3)-symmetry of the setup.

The variables of the reduced system are q, denoting the angle between the coordinate vec-

tors of the two bodies, p, the symplectic conjugate of ˙

qand m1,m2,m3which have the physical

meaning of the angular momentum components in the body frame.

In order to simplify our computations, we make the standard choice of an attracting potential

V(q)= −µ1µ2cot(q) describing the interaction of the particles; additionally, we demand that the

two particles be identical: via scaling their masses µ1and µ2can both be assumed to be 1. This

allows to write the system in a more palatable way, and introducing ξ=cot(q) makes it into a

polynomial one, guaranteeing the smoothness of the solution curves.

Collision trajectories

The reduced system has singularities of two types: antipodal and collisions, corresponding to the

cases q=πand q=0, respectively. We demonstrate that, for two equal masses, no trajectories

leading to antipodal collisions exist (the same proof can be immediately adapted to the case of

two points with arbitrary positive masses µ1µ2, see Remark 3.3). Thus, we are left with the task of

investigating collisions and near collision orbits.

To accomplish this, we use a variety of asymptotic estimates. In a neighbourhood of a point

t∗∈R∪{±∞}we say that f≺gif f/g→0 when t→t∗,f≻gif f/g→ +∞ and f∼gif f=hg

for some bounded function hthat does not tend to 0. Additional relations ≼and ≽are the logical

unions of respectively ≺and ≻with ∼. We will comment on these asymptotic estimates later on.

From the very deﬁnition of collision trajectories, we have that q→0+along a collision trajec-

tory, thus making ξ=cot(q)→+∞. Henceforth, ξwill act the ’measuring device’ for the growth of

the rest of the dynamical variables.

Leveraging on the form of the common level set of the Hamiltonian function Hand the Casimir

function C, we conclude that m3→0 and the growth of |p|does not exceed that of pξ.

Introducing the quantity I=〈q1−q2,q1−q2〉, we demonstrate that the collision time is ﬁnite,

as well as that ˙

q, together with p, tend to −∞ on collision trajectories.

Further assessment on the growth of pyields that p∼ −pξ, which, in turn, allows us to esti-

mate m3≼1

ξ. Using this, we show that m2−2m3ξ≼1

pξ, leading to a computation proving that the

projection of any collision trajectory on the sphere m2

1+m2

2+m2

3=Chas ﬁnite length (this sphere

is precisely the level set of the Casimir function).

Our main result concerning the ω-limit set for initial data that lead to collisions is Theorem

3.17, stating that the ω-limit set of collision trajectories is a singleton of the form {m1=m∗

1,m2=

m∗

2,m3=m∗

3,q=0, p=−∞}.

Lastly, we deduce that the global behaviour of the two bodies is as follows: as they approach

each other, as a pair they move towards the plane orthogonal to the vector of angular momentum.

Additionally, bodies cannot rotate around each other inﬁnitely many times.

Regularisation

The form of the reduced equations (3.1) suggests that the two-dimensional plane m2=m3=0,

m1= ±pCis invariant under the dynamics. The corresponding motion is the one where both

bodies move along the great circle that lies in the plane orthogonal to the vector of the angular

momentum. We refer to this setup as the motion in the invariant plane. Unless the bodies are in a

state of a relative equilibrium on the opposite sides of the sphere (an antipodal singularity that we

don’t consider), a collision is inevitable. Additionally, this case illustrates that, in contrast with the

Kepler problem, collisions can happen at any values of the momentum.

This system reduces to a two-dimensional one, and through a change of variables and intro-

duction of a ﬁctitious time we regularise the equations. The only equilibrium of the new system is

the origin, and we use non-homogeneous blow-up to investigate the trajectories nearby.

For the general case, we observe that after an appropriate change of variables, the equilibria

of the system form a two-dimensional plane; this entails that we only need to perform a three-

dimensional nonhomogeneous blow-up, as opposed to a ﬁve-dimensional one, which simpliﬁes

the problem greatly.

Thus two of the eigenvalues of the linearised matrix at newly obtained equilibria will be identi-

cally zero, but this is inconsequential, since the dynamics in the corresponding directions is trivial

(a plane of equilibria). However, since in the blow-up we perform the exceptional divisor is a two

dimensional sphere, we need to utilise two charts to cover it completely. This leads us to use two

different non-homogenous blow-ups in order to provide two charts.

We determine the number and types of equilibria on both charts and demonstrate that equi-

libria of the same type, when twice covered, correspond to the same directions in the original vari-

ables. Lastly, the dynamics of all variables near the exceptional divisor are described.

Acknowledgments The authors would like to thank the anonymous referee for fruitful sugges-

tions and having helped to improve the readability of the paper.

x′′

y′

−z′

x′

µ2

π−θ

µ1

Figure 2.1: Euler angles and the body frame. Notation −z′signiﬁes that the

z′-axis of the moving system points in the opposite direction.

2 Setup

The equations of motion of two free bodies on the sphere with masses µ1and µ2that act on one

another with an attracting force have the form ([10], [11])











q1=p1

µ1;

q2=p2

µ2;

p1=V¡||q1−q2||¢q1−〈p1,p1〉

µ1q1;

p2=V¡||q1−q2||¢q2−〈p2,p2〉

µ2q2;

||q1||=||q2||=1;

p1·q1=p2·q2=0.

(2.1)

with the energy function (Hamiltonian) Hgiven by

H=||p1||2

2µ1+||p2||2

2µ2+V(||q1−q2||). (2.2)

The problem has obvious SO(3)-symmetry, and, consequently, at least three conserved quantities:

the components of the total angular momentum L, of explicit form

L=q1×p1=q2×p2.

These quantities are the components of the equivariant momentum map.

We assume that the potential V¡||q1−q2||¢is an attracting one, meaning that V¡||q1−q2||¢→

−∞ when q1→q2and V¡||q1−q2||¢→+∞ when q1→−q2.

Such a choice of the potential immediately entails that the phase space of this problem is Q:=

T∗(S2×S2\∆) (taken with the standard cotangent bundle symplectic structure), where ∆, referred

to in the literature as ’the big diagonal’ is the set consisting of pairs of the same or antipodal points

on the sphere.

However, Qis 8-dimensional and equations (2.1) are a rather complicated system. In order

to simplify the problem, one might use symplectic reduction with respect to the aforementioned

SO(3)-symmetry of the problem. This reduction was accomplished by Borisov, Mamaev, Montaldi

and García-Naranjo in [5]. We give a brief description of if below; for more details and uses of this

method, see [5], [14] and [4].

To ’separate’ the symmetric action of the group, we re-parameterise the phase space. We intro-

duce the new ’body frame’ with axes x′,y′,z′(x,y,zdenoting the old coordinate axes ) that moves

with the two bodies in the following way: the coordinate vector of the ﬁrst body always coincides

with the z′-axis of the new system, and the basis changes in such a way that the second body is

always in z′−y′plane. We denote by qthe angle between the two coordinate vectors and with-

out loss of generality we may assume that the initial positions of the two masses are, respectively,

x1=(0,0,−1) and x2=(0,sin(q), −cos(q)).

As we have stated, the form of our potential prohibits collisions as well as the two bodies from

occupying antipodal points on the sphere. Hence, q∈I:=(0, π). We suppose that g∈SO(3) and

ζ∈T SO(3).

Since T SO(3) can be trivialised as SO(3) ×so(3), we can write

T I ×T SO(3) →TQ

(q,˙

q,θ,φ,ψ,ω1,ω2,ω3)7→(g·x1,g·x2,gζ·x1,gζ·x2+gx′

2˙

q), (2.3)

with gexpressed through the three Euler angles as

g=



cos(φ)cos(ψ)−cos(θ) sin(ψ)sin(φ)−sin(φ)cos(ψ)−cos(θ)sin(ψ)cos(φ) sin(θ)sin(ψ)

cos(φ)sin(ψ)+cos(θ) cos(ψ)sin(φ)−sin(φ)sin(ψ)+cos(θ)cos(ψ)cos(φ)−sin(θ)cos(ψ)

sin(θ)sin(φ) sin(θ) cos(φ) cos(θ)

, (2.4)

see Figure 2.1 for illustration. The variable ζhas the physical meaning of angular velocity in the

body frame ([18]). After this change the Hamiltonian turns into

H=1

2µ1µ2³µ2¡(m1−p)2+m2

2¢+m3¡−2µ2m2cot(q)+µ1m3csc2q+µ2m3cot2q¢´+

+p2

2µ2+V(q).

(2.5)

The non-zero Poisson brackets in the reduced variables are given by

{m1,m2}=−m3, {m2,m3}=−m1,

{m1,m3}=m2, {q,p}=1. (2.6)

The three invariant values ’collapse’ into one Casimir function

C=m2

1+m2

2+m2

3. (2.7)

For the purposes of this work, we choose V(q)= −µ1µ2cot(q), where qis the angle between

the two coordinate vectors of the particles. This is a standard choice of potential for the N-body

problem on a sphere ([6]) (and its hyperbolic analogue, on a Lobachevsky plane), arising as well as

one of the solutions of the generalized Bertrand problem of ﬁnding potentials depending only on

geodesic distance and with closed orbits [16].

With these assumptions made, the ﬁve-dimensional equations of motion have the form











m1=1

µ1µ2(−m2m3µ2+µ2cot(q)(−m2

2+m2

3+m2m3cot(q)) +m2m3µ1csc(q)2)

m2=1

(µ1µ2)(m3(2m1−p)µ2+m1m2µ2cot(q)−m1m3(µ1+µ2) csc(q)2)

m3=1

µ1(m2p−m1m3cot(q))

q=1

µ1µ2(−m1µ2+p(µ1+µ2))

p=1

µ1µ2((−µ2(m2m3+µ2

1µ2)+m2

3(µ1+µ2)cot(q)) csc(q)2)

(2.8)

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Collisiontrajectoriesandregularisationoftwo-bodyproblemonS2AlessandroArsie&NataliyaA.BalabanovaAugust2022AbstractInthispaper,weinvestigatecollisionorbitsoftwoidenticalbodiesplacedonthesurfaceofatwo-dimensionalsphereandinteractingviaanattractingpotentialoftheformV(q)=−cot(q),whereqistheangleformedbyt...

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Collision trajectories and regularisation of two-body problem onS2 Alessandro Arsie Nataliya A. Balabanova

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