## ABSTRACT

This research has investigated an improved version of the classic restricted three-body

problem where both primaries are considered as non-spheroids as well as sources of

radiation, and are enclosed by a belt of homogeneous cluster of material points centered at

the mass center of the system. It studied the effects of the gravitational potential created by

the belt on the existence of libration points and their linear stability when the primaries

are:(a) radiating-oblate and (b) radiating-triaxial bodies. To this end, the equations that

govern the motion of the infinitesimal body have been derived, and the positions of the

libration points areobtained. It has been established that the equations and the positions of

the libration points are affected by the aforementioned perturbations, such that in addition

to the usual five (three collinear, two triangular) libration points of the classicrestricted

three-body problem, there exist four new collinear points in the case of radiating-oblate

primaries and up to two in the case of radiating-triaxial primaries. The triangular points are

seen to be stable for 0 c and unstable for

1

,

2 c where c is the critical mass

parameter influenced by the radiation and oblateness/triaxiality of the primaries and

potential from the belt. The combined effect of these perturbations is a stabilizing tendency

when the primaries are oblate and adestabilizing tendency when they are triaxial. The

classical three collinear libration points 1, 2 3 (L L , L ) remain unstable, whereas some of

the new libration points have been found to be stable. Furthermore, periodic orbits around

the stable triangular points owing to the amalgamated effect of the radiation and oblateness

up to the coefficient J4 of the primaries and potential from the belt are found to be ellipses.

vii

## TABLE OF CONTENTS

TITLE PAGE ………………………………………………………………………………………………………….. i

DECLARATION …………………………………………………………………………………………………….. ii

CERTIFICATION …………………………………………………………………………………………………..iii

DEDICATION ……………………………………………………………………………………………………….. iv

ACKNOWLEDGMENTS ………………………………………………………………………………………… v

ABSTRACT…………………………………………………………………………………………………………… vi

TABLE OF CONTENTS………………………………………………………………………………………… vii

LIST OF FIGURES ……………………………………………………………………………………………….. xii

LIST OF TABLES ………………………………………………………………………………………………… xiv

DEFINITIONS OF TERMS …………………………………………………………………………………… xvi

LIST OF SYMBOLS ……………………………………………………………………………………………. xvii

CHAPTER ONE ……………………………………………………………………………………………………… 1

1.0 GENERAL INTRODUCTION…………………………………………………………………………….. 1

1.1 Introduction ……………………………………………………………………………………………………….. 1

1.2Statement of the Problem ……………………………………………………………………………………… 2

1.3 Justification of the Study …………………………………………………………………………………….. 3

1.4Aim and Objectives of the Study …………………………………………………………………………… 4

1.5 Research Problems …………………………………………………………………………………………….. 4

viii

1.6 Methodology …………………………………………………………………………………………………….. 5

1.7 Theoretical Framework ……………………………………………………………………………………….. 6

1.7.1 Circular restricted three-body problem ……………………………………………………………. 6

1.7.2 Ellipsoid ……………………………………………………………………………………………………. 9

1.7.3 Radiation …………………………………………………………………………………………………… 11

1.7.4 Potential of a belt ……………………………………………………………………………………….. 11

1.8 Some Basic Concepts………………………………………………………………………………………… 12

1.8.1 Velocity in the rotating frame ………………………………………………………………………. 12

1.8.2 Hamiltonian equations of motion ………………………………………………………………….. 14

1.8.3 Lyapunov stability ………………………………………………………………………………………. 16

CHAPTER TWO …………………………………………………………………………………………………… 18

2.0LITERATURE REVIEW …………………………………………………………………………………… 19

2.1Introduction ………………………………………………………………………………………………………. 19

2.2 The Restricted Three-Body Problem …………………………………………………………………… 19

2.3 The Restricted Three-Body Problem with Radiation …………………………………………….. 20

2.4 The Restricted Three-Body Problem with Asphericity ………………………………………….. 21

2.5 The Restricted Three-Body Problem with Belt …………………………………………………….. 23

2.6 Periodic Orbits in the Restricted Three-Body Problem ………………………………………….. 24

CHAPTER THREE ……………………………………………………………………………………………….. 27

3.0 OBLATENESS AND RADIATION OF THE PRIMARIES WITH POTENTIAL FROM A BELT ………………………………………………………………………………… 28

ix

3.1 Introduction ……………………………………………………………………………………………………… 28

3.2Derivation of Equations of Motion ………………………………………………………………………. 28

3.2.1 Mathematical formulations of the problem …………………………………………………….. 28

3.2.2 The kinetic energy ………………………………………………………………………………………. 29

3.2.3 The potential energy ……………………………………………………………………………………. 30

3.2.4 Equations of motion in dimensional variables ………………………………………………… 31

3.2.5 The mean motion n……………………………………………………………………………………… 38

3.2.6 Equations of motion in dimensionless variables ……………………………………………… 39

3.3 The Jacobian Integral and Zero Velocity Surfaces ………………………………………………… 42

3.3.1 The Jacobian integral ………………………………………………………………………………….. 42

3.3.2 Zero velocity surfaces …………………………………………………………………………………. 43

3.4Locations of Libration Points ……………………………………………………………………………… 46

3.4.1 Triangular libration points …………………………………………………………………………… 47

3.4.2 Locations of collinear libration points …………………………………………………………… 50

3.5Stability of Libration Points ………………………………………………………………………………… 55

3.5.1 Variational equations ………………………………………………………………………………….. 56

3.5.2 Characteristic equation ………………………………………………………………………………… 61

3.5.3 Stability of triangular libration points ……………………………………………………………. 63

3.5.4 Stability of collinear libration points …………………………………………………………….. 74

3.6Periodic Orbits around the Triangular Libration Points ………………………………………….. 80

x

3.6.1 The frequencies of the periodic orbits ……………………………………………………………. 80

3.6.2 The nature of periodic orbits ………………………………………………………………………… 89

CHAPTER FOUR………………………………………………………………………………………………… 126

4.0TRIAXIALITY AND RADIATION OF THE PRIMARIES WITH POTENTIAL FROM A BELT …………………………………………………………………………………………………… 126

4.1Introduction …………………………………………………………………………………………………….. 126

4.2Derivation of Equations of Motion …………………………………………………………………….. 126

4.2.1 Mathematical formulations of the problem …………………………………………………… 126

4.2.2 Energy of the infinitesimal mass …………………………………………………………………. 126

4.2.3 Equations of motion in dimensional variables ………………………………………………. 127

4.2.4 Equations of motion in dimensionless variables ……………………………………………. 129

4.2.5 The mean motion n……………………………………………………………………………………. 135

4.3Location of Libration Points ……………………………………………………………………………… 137

4.3.1 Location of triangular libration points …………………………………………………………. 139

4.3.2 Locations of collinear libration points …………………………………………………………. 143

4.4Stability of Libration Points. ……………………………………………………………………………… 146

4.4.1 Variational and characteristic equations ………………………………………………………. 146

4.4.2 Stability of triangular libration points. …………………………………………………………. 149

4.4.3 Stability of collinear libration points …………………………………………………………… 155

CHAPTER FIVE …………………………………………………………………………………………………. 160

5.0RESULTS AND DISCUSSIONS ………………………………………………………………………. 160

xi

5.1Introduction …………………………………………………………………………………………………….. 160

5.2 Oblateness and Radiation of the Primaries with Potential from a Belt …………………… 160

5.2.1 Equations of motion ………………………………………………………………………………….. 160

5.2.2 Locations of libration points ………………………………………………………………………. 161

5.2.3 Stability of libration points ………………………………………………………………………… 172

5.2.4 Periodic orbits around the triangular libration points …………………………………….. 176

5.3Triaxiality and Radiation of the Primaries with Potential from a Belt …………………….. 179

5.3.1 Equations of motion ………………………………………………………………………………….. 179

5.3.2 Locations of libration points ………………………………………………………………………. 179

5.3.3 Stability of libration points ………………………………………………………………………… 188

CHAPTER SIX ……………………………………………………………………………………………………. 194

6.0 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ………………………….. 194

6.1 Summary ……………………………………………………………………………………………………….. 194

6.2 Conclusions ……………………………………………………………………………………………………. 197

6.3 Recommendations …………………………………………………………………………………………… 199

REFERENCES ……………………………………………………………………………………………………. 200

xii

## CHAPTER ONE

1.0 GENERAL INTRODUCTION

1.1 Introduction

The study of the motion of three bodies of finite masses, moving under their mutual

gravitational attraction, is called the three-body problem (3BP). It has been the subject of

several studies for many years. However, the solution of the problem remains a formidable

challenge. The problem simplifies to the restricted three-body problem (R3BP) if one of the

bodies is assumed to have infinitesimal mass and does not, therefore, affect the motion of

the remaining two massive bodies, called the primaries. If further, the primaries move on

circular orbits relative to their barycenter, it is known as the circular restricted three-body

problem (CR3BP). Then, the goal in this CR3BP is the solution for the motion of the body

of infinitesimal mass. A complete analytic solution to the CR3BP does not exist;

nevertheless, libration (equilibrium) solutions satisfying precise conditions are obtainable.

Three libration solutions L1 , L2 , 3 L are collinear with the primaries; and two others, called

the triangular libration solutions 4 5 L , L , form equilateral triangles with the primaries. The

infinitesimal mass would remain fixed if placed with zero velocity at any of these libration

points. Thus, they provide ideal locations for space based telescopes and other space

science missions. The collinear points are unstable; while the triangular points are

conditionally stable. The CR3BP is a stimulating and active research field that has been

receiving considerable attention of mathematicians and astronomers since it was initially

considered by L. Euler in 1772 (Szebehely, 1967). The most evident reason for this

sustained interest is that the model of the circular restricted three-body problem can serve

as a first approximation in a number of real situations in astronomy (dynamics of the solar

2

and stellar systems, lunar theory and artificial satellites). A deeper motivation emanates most likely from the fact that no general solution exist, in spite of the apparent simplicity of the problem. The nearly circular motion of the planets about the sun and the small masses of asteroids and satellites of planets compared to the planets‟ masses, initially advocated the invention of the CR3BP.

1.2 Statement of the Problem

The classical CR3BP considers only the effects of the gravitational attraction on the infinitesimal body, but in reality there are other perturbing forces that affect the motion of the infinitesimal body. In the presence of a luminous body, it is altogether inadequate to consider merely the gravitational interaction force. Poynting (1903) pointed out that particles, such as small meteors or cosmic dust are comparably affected by the gravitation and light radiation force as they approach luminous celestial bodies. Radzievskii (1950) found that an allowance for direct solar radiation pressure results in a change in the positions of the libration points. There are rings of dust particles in the stellar systems which are regarded as the young analogues of the Kuiper belt (Greaves et al. 1998).Trilling et al. (2007) detected debris rings in many main-sequence stellar binary systems using the Spitzer Space Telescope. Out of an observed 69 A3-F8 main sequence binarystar systems nearly 42 (60%)showed dust rings surrounding binary stars. The gravitational potential due to these belts have great influence on the infinitesimal body (Jiang and Yeh, 2004a). In the classical CR3BP, the sphericity of the bodies is presumed whereas in reality numerous celestial bodies are non-spherical in shape. For instance, Jupiter and Saturn are oblate while Pluto and its moon Charon are triaxial. Singh and Ishwar (1999) revealed that lack of sphericity of the primaries affects the motion of the infinitesimal body.

3

Thus, the classical CR3BP does not adequately represent the motion of the infinitesimal body in the presence of any perturbing agent such as radiation pressure, non-sphericity of a body and gravitational potential from a belt. Therefore, the problem in this work is to modify the classical CR3BP to include the effects of radiation of the primaries, asphericity of the bodies and gravitational potential from a belt.

1.3 Justification of the Study

According to Henri Poincare: The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living (Poincare, 1958, p.8). Great minds throughout the ages have been fascinated in the celestial motions they observed in the night sky. The human species stands on the brink of a new frontier, the evolution from a planet-bound to a space-faring advancement. Just as the transition from hunter-gatherer to farmer necessitated new techniques, so the expansion into the space, in terms of dynamics of artificial satellite, requires the formulation of new models that include the effects of some of the perturbing forces on the satellite. More importantly, the study of effect of these forces on its stationary (libration) positions in the space; since they provide ideal locations for space based telescopes and other space science missions. This study is one of such models that incorporated the effects of radiation of the primaries, asphericity of the bodies and gravitational potential from a belt on the satellite.

4

1.4 Aim and Objectives of the Study

The aim of this research is to examine the existence of libration points and their stability behaviors under the combined effect of radiation, asphericity of the primaries and gravitational potential from the belt within the framework of the restricted three-body problem. Moreover, the aim will be achieved through the following specified objectives, which are to:

i. derive equations of motion of the infinitesimal body in the modified restricted three-body problem.

ii. establish the existence of libration points in the modified restricted three-body problem.

iii. analyze the effects of the modification of the restricted three-body problem on the locations of the libration points.

iv. investigate the linear stability of the libration points in the modified restricted three-body problem.

v. determine periodic orbits about triangular libration points in the modified restricted three-body problem.

1.5 Research Problems

The research problems to be addressed by the study are:

i. Influence of zonal harmonics (oblateness up to J4) and potential from a belt on the stability of libration points in the CR3BP.

ii. Combined effect of zonal harmonics, radiation and potential from a belt on the stability of libration points in the CR3BP.

5

iii. Effects of triaxiality and potential from a belt on the stability of libration points in the CR3BP.

iv. Combined effect of triaxiality,radiation and potential from a belton the stability of libration points in the CR3BP.

v. Periodic orbits around the libration points in photogravitationalCR3BP with zonal harmonics and potential from the belt.

1.6 Methodology

The methodology employed in this investigation involves:

i. Extension and modification of existing mathematical models of the CR3BP (Szebehely , 1967; Singh and Taura 2013).

ii. Obtaining analytical solutions by considering only linear terms in very small quantities (quantities strictly very less than one), in all analytical computations.

iii. Validating the obtained results by comparing them with well-established results of CR3BP.

iv. Studying the effects of the perturbations analytically and numerically on the obtained results.

v. Examining periodic orbits emanating from the triangular libration solutions obtained.

vi. Interpretation, discussion and drawing conclusions.

6

1.7 Theoretical Framework

The outlines of the theoretical bases on which the problem is built are first presented:

1.7.1 Circular restricted three-body problem

It describes the motion of an infinitesimal mass moving under the gravitational influence of

two bodies, called primaries which move in circular orbits around their center of mass on

account of their mutual attraction and the infinitesimal mass not affecting the motion of the

primaries. The primaries are fixed on the x-axis in the coordinate system (oxyz) rotating at

an orbital angular velocity n, with the origin (axis of rotation) at the centre of their masses.

The (x, y) plane is the plane of motion of the primaries and the z-axis is orthogonal to the

(x, y) plane.

In the circular restricted three body problem, the units are chosen in such a way that the

properties of the system depend on a single parameter.

i. The sum of the masses of the primaries is taken as the unit of mass. The mass of the

smaller primary is denoted by μ, whence the mass of the bigger primary 1-μ, where

2

1 2

m

m m

is the mass parameter and m1, 2 m are the masses of the bigger and

smaller primaries, respectively.

ii. The distance between the primaries is a unity. The distances of the smaller primary

and bigger primary from the centre of mass are then 1 − μ and μ, respectively.

7

iii. The unit of time is chosen so that the mean motion of the primariesn is equal to1.

From these it follows that the gravitational constant is unity. The only remaining

parameter is μ.

Then, the equations of motion of the infinitesimal mass m with the coordinates x, y, z

relative to the frame (Figure 1.1) that rotates with the primaries are:

2 ,

2 ,

x

y

z

x y

y x

z

(1.1)

where x, y, zis

2 2

1 2

1 1

2

x y

r r

with 2 2 2 2

1 r x y z , 2 2 2 2

2 r x 1 y z , r1, r2are the distances of the

infinitesimal mass m from the bigger primary 1 m and smaller primary 2 m , respectively.

Figure 1.1: Planer view of the rotating frame of reference.

x

y

x

y

1 m

2 m

2

m(x, y, z)

n 1

1 r

2 r

8

A complete analytic solution of the CR3BP does not exist; nevertheless, libration

(equilibrium) solutions satisfying precise condition are obtainable. Three libration solutions

L1 , L2 , 3 L are collinear with the primaries; and two others, called the triangular libration

solutions 4 5 L , L , form equilateral triangles with the primaries. 4 L and 5 L located at

(1/ 2 , 3 / 2) in the xy-plane. The infinitesimal mass would remain fixed if placed

with zero velocity at any of these libration points. The collinear points are unstable; while

the triangular points are conditionally stable.If the mass ratio 0.25, for example, we

have the coordinates of the libration points

1 2 3 4 5 L (1.2659,0), L (0.3607,0), L (-1.1032,0) , L (0.25,0.8660), L (0.25, 0.8660) as

shown in Figure 1.2.

Figure 1.2: Locations of libration points in the CR3BP when 0.25.

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

m

1

m

2

L

4

L

5

L

3

L

2

L

1

Locations of the libration points when the mass ratio is 0.25

x

y

9

1.7.2 Ellipsoid

An ellipsoid is a closed second-order surface; it is a three-dimensional equivalent of an

ellipse. The canonical equation of an ellipsoid in the Cartesian coordinate system has the

form

2 2 2

2 2 2 1,

x y z

a b c

where a and b are the equatorial radii (along x and y axes) and c is the polar radius,

determining the shape of the ellipsoid. If

a b c : triaxial ellipsoid;

a b c : oblate ellipsoid (oblate spheroid);

a b c : prolate ellipsoid;

a b c : sphere ellipsoid( spheroid).

1.7.2.1 Potential of an oblate body

Peter and Lissauer (2007) showed that, in free space, the gravitational potential exterior to

an oblate body with its mass distributed symmetrically about its equator can be expanded in

terms of Legendre polynomials in the form:

2

2 2

1

( , , ) 1 (cos ) .

n

o o

o o n n

o n o

Gm R

V r J P

r r

(1.2) This is

expressed in standard spherical coordinates, with the longitude and representing the

angle between the body‟s symmetry axis and the vector to a particle ro (i.e., the

colatitudes). Ro is the mean radius of the body.

10

J2n are the gravitational moments determined by the body‟s mass distribution. They are

called zonal harmonic coefficients. Higher order zonal harmonics are very small compared

to J2.

The terms P2n (cos) are the Legendre polynomials, given by

2

2

2

2 2 2

1

( ) 1 .

2 (2 )!

n

n

n n n

d

P x x

n dx

(1.3)

Now, if the particle is moving in a circular orbit in the equatorial plane (=900) the

potential (Equation (1.2)) becomes

2 4 6

2 4 6

3 5 7

1 3 5

( , , ) …

2 8 16

o o o

o o o

o o o o

J R J R J R

V r Gm

r r r r

(1.4)

1.7.2.2 Potential of a triaxial body

The gravitational potential due to a triaxial body of mass t m to an external particle is given

by (McCuskii, 1963)

2 1 2 3

1

1 3 ,

2

t

t

t t t

Gm

V I I I I

r m r

(1.5)

where

2 2 2 2 2 2

1 2 3 , , ,

5 5 5

t t t b c m c a m b a m

I I I

are the principal moments of

inertia of t m about its centre of mass,

2 2 2

1 2 3 I I l I m I n is the moment of inertia about the line joining the centre of the mass

of the body t m and the particle. l, m,n are the direction cosines of the vector from the

particle relative to the principle axes of . t m

11

1.7.3 Radiation

Radzievskii (1950) formulated that the effect of radiation pressure of a source on a small

particle, is expressed by means of a reduction factor q = 1 − p. Parameter p, that is often

mentioned as the radiation coefficient, is the ratio of force r F which is caused by radiation to

force g F which results from gravitation, that is r

g

F

p

F

. Since in most cases the

gravitational force exceeds radiation, we shall consider that q 0 and consequently

0 p 1. This formulation is based on some assumptions that can be summarized as

follows:

i. The radiation emitted from the primaries (stars) influences the motion of the small

body but does not affect the motion of the others.

ii. The mass and the temperature of the radiating star are almost constant.

iii. Light has no fluctuations, it propagates in straight lines and screening or shadowing

effects produced by other bodies of the system are negligible.

iv. The force of light pressure is inversely proportional to the square of the distance

between the particle and the illuminating body.

1.7.4 Potential of a belt

The gravitational potential from a belt (circular cluster of material points) centered at the

origin of coordinates system oxyz, as specified by Miyamoto and Nagai (1975) is

2

2 2 2

( , ) , b M

V r z

r a z b

(1.6)

12

where b M is the total mass of the belt, r2 x2 y2 , a and b are parameters which

determine the density profile of the belt. The parameter a controls the flatness of the profile

and is known as the flatness parameter. The parameter b controls the size of the core of the

density profile and is called the core parameter. When a = b = 0, the potential reduces to

the one by a point mass. Restricting ourselves to the xy plane , Equation (1.6) becomes

1 2

2 2

,0 , b M

V r

r T

where T a b. (1.7)

1.8 Some Basic Concepts

Fundamental ideas which are crucial to the study are velocity in the rotating frame,

Hamiltonian equations of motion and Lyapunov stability.

1.8.1 Velocity in the rotating frame

Let OX, OY, OZ constitute a rectangular frame, the point O being fixed for the moment,

and the frame rotating instantaneously about an axis through O with angular velocity

1 2 3

iˆ ˆj kˆ,

as shown in Figure 1.3.

Z

13

3 (1,2 ,3 )

P(x, y, z)

kˆ

O ˆj

ˆi

2

Y

1

X

Figure 1.3:

Rotating coordinate frame

P(x, y, z) is any point in the space which can move relative to the frame with a position

vector r xiˆ yˆj zkˆ,

then the velocity of P is

ˆ ˆ ˆ

ˆ ˆ ˆ .

di dj dk

r xi x yj y zk z

dt dt dt

(1.8)

Now, since the axes are rotating

ˆ ˆ ˆ

, ,

di dj dk

dt dt dt

are non-zero.

diˆ

dt

is the velocity of the point

(1,0,0), this is due to the spins of 2 about OY and 3 about OZ. That is the angular

velocity components rotate the system around unit axis, so that for example, 2 tends to

rotate iˆ towards the kˆ direction and 3 tends to rotate ˆi towards the ˆj

direction. Thus,

3 2

ˆ

ˆ ˆ,

di

j k

dt

and we have

14

3 2 1 3 2 1

r xiˆ yˆj zkˆ ( ˆj kˆ)x ( kˆ iˆ)y ( iˆ ˆj)z

.

Or

r xiˆ yˆj zkˆ r

. (1.9)

1.8.2 Hamiltonian equations of motion

A system with s degrees of freedom possesses s Lagrange‟s equations of motion of the

form

0

i i

L d L

q dt q

, (1.10)

where L=kinetic energy(K.E)- potential energy(V) of the system , , ( 1,2,3…, ) i i q and q i s

are the generalized co-ordinates and generalized velocities respectively and t is time. Thus

the Lagrangian formulation is a description of mechanics in terms of the generalized coordinates

and generalized velocities with time t as a parameter. Hamilton‟s formulation is

an alternative to the Lagrangian formulation in which the independent variables are

generalized coordinates i q and momenta i p defined as

( , , )

. i i

i

i

L q q t

p

q

(1.11)

The alteration in basis from the set , , , , i i i i q q t to q p t is achieved by

Legendretransformation.

The Hamiltonian function

We start by expressing the differential of the Lagrangian ( , , ) i i L q q t , as

i i

i i i i

L L L

dL dq dq dt

q q t

.

15

Or

, i i i

i i i

L L

dL dq p dq dt

q t

(1.12)

where

. i

i

L

p

q

(1.13)

Using Equation (1.13) in Equation (1.10), yields

, i

i

L

p

q

so that Equation (1.12) can be rewritten as

. i i i i

i i

L

dL p dq p dq dt

t

(1.14)

The classical Hamiltonian function ( , , ) i i H q p t , is generated by the

Legendretransformation

( , , ) ( , , ), i i i i i i

i

H H q p t p q L q q t

(1.15)

which has the differential

. i i i i

i i

dH q d p p dq dL

(1.16)

Substituting Equation (1.14) in Equation (1.16), we have

i i i i

i i

L

dH q dp p dq dt

t

. (1.17)

Since dH can also be expressed as

, i i

i i i i

H H H

dH dq dp dt

q p t

(1.18)

by comparing Equations (1.17) and (1.18), we obtain the relations

16

,

,

i

i

i

i

H

q

p

H

p

q

(1.19)

.

H L

t t

(1.20)

The equations of Equation (1.19) are known as Hamilton‟s canonical equations. They form

a set of 2s first-order equations of motion substituting the s second order Lagrange‟s

equations of motion.

1.8.3 Lyapunov stability

Consider the systems of autonomous ordinary differential equations, written in vector form

as x f (x) , that have an isolated equilibrium point x x at the origin. Then, * x is said to

be stable in the sense of Lyapunov orLyapunov stableif for all 0 there exists a δ >0

(depending on ε) such that for all 0 x(t ) with 0 x(t ) x , implies x(t) x for all

t 0, where 0 x(t ) represents the system at some initial time 0 t . That is x* is Lyapunov

stable if for any ε-neighborhood of x* a smaller – neighborhood can be found such that if

the system is perturbed within the -neighborhood, it never leaves the ε-neighborhood.

If x* Lyapunov stable and x(t) x 0 as t , then x* is called an asymptotically

stable equilibrium point. That is, a solution that starts sufficiently close to an asymptotically

stable equilibrium point will not only remain nearby, but will actually be “attracted” by it,

as illustrated in Figure 1.4. If x*is not Lyapunov stable, it is said to be unstable in the sense

of Lyapunov.

Asymptotic stable

Stable

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Figure 1.4: Graphical representation of Lyapunov stability. Stability of an equilibrium point One of the techniques to determine stability of an equilibrium x*of a system of differential equations is to analyze the linearized system around it. Then, the solutions to linearized differential equations with constant coefficients can be expressed as linear combination of the product of a polynomial and an exponential function. Therefore solving a linear differential equation with constant coefficients can be transformed to solving a linear algebraic problem. Stability or instability then may follow from the roots of the characteristic equation of the linearized system.

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Lyapunov proved that an equilibrium point x* is stable if all the roots of the characteristic equation of the linearized system are either negative real numbers or distinct pure imaginary numbers or real parts of the complex numbers are negative.