Chinmoy Taraphdar - The Classical Mechanics (2007)

The Classical Mechanics

Chinmoy Taraphdar

Asian Books Private Limited

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The Classical

Mechanics

Chinmoy Taraphdar Lecturer, Dept. of Physics

Bankura Christian College

Bankura, West Bengal

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I st Published 2007

ISBN 978-81-8412-039-4

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Preface

This book, organised into ten chapters, is written to cover the syllabus of classical

mechanics for the students of physics at the graduate and postgraduate level. I

hope that the clear, lucid and comprehensive coverage of this book will help

students to gain a thorough grounding of the subject.

The beginning part of this book explains several chapters on the basis of

Newtonian mechanics and then the text explains the generalised co-ordinates

and Lagrangian mechanics along with Hamiltonian mechanics basically for the

holonomic system. The concluding chapter deals with the canonical

transformations by which the solution to the particular problem can be obtained

trivially. Some problems have been worked out to aid in understanding the

underlying theory at the end of each chapter.

Finally, the expressions of acknowledgements. I am indebted to my

colleagues, students who have kindly given me valuable comments and

suggestions. I acknowledge with a deep sense of gratitude my indebtedness to

the authors whose standard works in the field I have freely consulted to my

benefit. I also acknowledge my indebtedness to my wife 'Anamika', my daughter

'Sreetama' and my son 'Jyotirmoy' for their help at every stage of the preparation

of the manuscript. My special thanks are due to all concerned of 'Asian Books

Private Limited', especially to Ms. Purobi Biswas, Production Manager. Mr.

Subhadip Khan, the Branch Manager of Kolkata office for their kind help in

bringing out the volume in its admirable form and bearing with me at every

stage with unfailing patience and good humour.

It is fervently hoped that the book will be of value to the students and

teachers alike. Comments and suggestions for improvements to the text will be

thankfully acknowledged.

Chinmoy Taraphdar

Contents

Preface

Chapter 1. Vector

1.1. Fundamental Concept of Scalar and Vectors, 1

1.2. Unit Vectors and General representation of a vector, 1

1.3. Multiplication and Division of Vectors by Scalar, 2

1.4. Collinear Vectors, 3

1.5. Linear Dependence or Independence of Vectors, 3

1.6. Addition and Subtraction of two Vectors, 4

1.7. Addition of More Than Two Vectors, 5

1.8. Position Vector and Its Representation in Co-ordinate System, 5

1.9. Condition of Co-planarity of Vectors, 7

1.10. Rotational Invarience of Vector in Reference Frame, 8

1.11. Product of Two Vectors, 8

1.12. Scalar Tripple Product, 10

1.13. Vector Tripple Product, 11

1.14. Pseudo Vectors and Pseudo Scalars, 11

1.15. Vector Derivatives (Ordinary), 13

1.16. Vector Derivatives (Partial) and Vector Operators, 13

1.17. Laplacian and D'Alambertian Operator, 18

1.18. Vector Integration, 19

1.19. Gauss's Divergence Theorem, 20

1.20. Green's Theorem, 21

1.21. Stoke's Theorem, 22

1.23. Reciprocal Vectors, 24

1.23. Scalar and Vector Field, 24

1.25. Elementary Idea about Vector Space, 24

1.25. Linear Operator in Vector Space, 25

Summary, 26

Worked Out Examples, 29

Exercises, 34

Chapter 2. Linear Motion

2.1. Introduction, 37

2.2. Kinematics, 37

2.3. Basic Definitions of Required Parameters, 37

2.4. Velocity and Acceleration in Several Co-ordinate System, 40

2.5. Tangential and Normal Component of Velocity and Acceleration, 41

2.6. Radial and Transverse Component of Velocity and Acceleration, 42

2.7. Newton's Laws of Motion, 44

2.8. Accelerated Linear Motion, 45

(iii)

1-36

37-64

(vi)

2.9. Graphical Treatment of Linear Motion, 45

2.10. Conservation of Linear Motion, 46

2.11. Time Integral of Force (Impulse), 46

2.12. Work,47

2.13. Power, 48

2.14. Energy: Kinetic and Potential, 48

2.15. Conservative Force, 49

2.16. Conservation of Energy, 49

2.17. Center of Mass and Its Motion, 50

2.18. The Two Body Problem, 51

2.19. Application of the Principle of Linear Motion, 52

2.20. Mechanics of Variable Mass, 53

Summary, 55

Worked Out Examples, 58

Exercises, 64

Chapter 3. Rotational Motion: Rigid Body Rotation 65-94

3.1. Introduction, 65

3.2. Angular velocity and Angular Momentum, 65

3.3. Angular Acceleration, 66

3.4. Moment of Inertia and Torque, 67

3.5. Centrifugal force, 70

3.6. Rotational Kinetic Energy, 71

3.7. Angular Momentum for Rigid Body Rotation, 72

3.8. Kinetic Energy for Rigid Body Rotation, 74

3.9. Axes theorem for Moment of Inertia, 75

3.10. Calculation of Moment of Inertia in different cases, 76

3.11. Momental Ellipsoid or Ellipsoid of Inertia, 83

3.13. Moment and product of Inertia and Ellipsoid of inertia of some,

symmetrical bodies, 84

3.12. Moment of Inertia Tensor, 87

3.14. Routh's Rule, 87

3.15. Euler's Angles, 88

Summary,89

Worked Out Examples, 91

Exercises, 93

Chapter 4. Reference Frame 95-111

4.1. Introduction, 95

4.2. Non Inertial Frame and Pseudo Force, 95

4.3. Effect of rotation of earth on acceleration due to gravity, 99

4.4. Effect of Coriolis Force on a particle moving on the surface of earth, 101

4.5. Effect of Coriolis force on a particle falling freely under gravity, 103

4.6. Principle of Foucault's Pendulum, 104

4.7. Flow of River on Earth Surface, 106

Summary, 106

Worked out Examples, 107

Exercises, 110

(vii)

Chapter 5. Central Force 112-124

5.1. Introduction, 112

5.2. Definition and Characteristics of Central force, 112

5.3. Conservation of Angular Momentum under Central Force, 113

5.4. Conservation of energy under central force, 113

5.5. Equation of motion under attractive central force, 115

5.6. Application of central force theory to gravitation Deduction of

Keplar's law, 116

5.7. Energy conservation for planetory motion, 118

5.8. Stability of Orbits, 120

Summary, 120

Worked out Examples, 121

Exercises, 124

Chapter 6. Theory of Collision 125-137

6.1. Introduction, 125

6.2. Characteristics of Collision, 125

6.3. Lab Frame and Center of Mass Frame, 126

6.4. Direct or Linear Collision, 127

6.5. Characteristic of Direct Collision, 129

6.6. Maximum Energy transfer due to head on elastic collision, 130

6.7. Oblique Collision, 131

Summary, 132

Worked Out Examples, 133

Exercises, 137

Chapter 7. Conservation Principle and Constrained Motion 138-154

7.1. Characteristics of Conservation Principle, 138

7.2. Mechanics of a single particle and system of particles, 139

7.3. Conservation of linear momentum, 140

7.4. Conservation of Angular Momentum, 141

7.5. Conservation of Energy, 142

7.6. Constrained Motion, 145

7.7. Generalised Co-ordinates and other Generalised Parameters, 146

7.8. Limitation of Newton's Law, 151

Summary, 151

Worked out examples, 153

Exercises, 154

Chapter 8. Variational Principle and Lagrangian Mechanics 155-193

8.1. Introduction, 155

8.2. Forces of Constraint, 155

8.3. Virtual Displacement, 156

8.4. Principle of Virtual Work, 156

8.5. D' Alembert's Principle, 157

8.6. Lagrange's equations for a holonomic System, 158

8.7. Lagrange's equation for a conservative, non-holonomic system, 160

8.8. Introduction to Calculus of variations, 161

8.9. Variational Technique for many independent variables:

Euler-Lagrange's differential equation, 165

( viii)

8.10. Hamilton's Variational Principle, 166

8.11. Derivation of Hamilton's principle from Lagrange's equation, 167

8.12. Derivation of Lagrange's equations from Hamilton's principle, 168

8.13. Derivation of Lagrange's equation from D' Alambert's principle, 169

8.14. Derivation of Hamilton's Principle from D' Alambert's Principle, 171

8.15. Cyclic or Ignorable Co-ordinates, 172

8.16. Conservation Theorems, 172

8.17. Gauge Function for Lagrangian, 175

8.18. Invarience of Lagrange's equations under Generalised

Co-ordinate, transformations, 177

8.19. Concept of Symmetry: Homogeneity and Isotropy, 178

8.20. Invarience of Lagrange's equation under Galilean Transformation, 179

8.21. Application's of Lagrange's equation of motion in several mechanical

systems, 180

Summary, 186

Worked Out Examples, 187

Exercises, 193

Chapter 9. Hamiltonian Formulation in Mechanics 194-209 9.1. Introduction, 194

9.2. Hamiltonian of the System, 194

9.3. Concept of Phase Space, 195

9.5. Hamilton's Canonical Equations in different Co-ordinate System, 197

9.6. Hamilton's Canonical equations from Hamilton's Intergral Principle, 199

9.7. Physical Significance of Hamiltonian of the System, 201

9.8. Advantage of Hamiltonian Approach, 201

9.9. Principle of Least Action, 201

9.10. Difference between Hamilton's Principle and the principle of least,

action, 203

9.11. Application of Hamilton's Canonical Equations, 203

Summary, 206

Worked Out Examples, 206

Exercises, 209

Chapter 10. Canonical Transformations 210-232

10.1. Introduction to Canonical Transformations, 210

10.2. Hamilton-Jocobi Method, 215

10.3. Application of Hamilton-Jacobi method to the particle falling freely, 217

10.4. Han,iIton's Characteristics function, 219

10.5. Action and Angle Variables, 220

10.6. Application of Action Angle Variables to Harmonic Oscillator Problem, 221

10.7. Poisson's Bracket, 222

10.8. Poisson's Theorem, 223

10.9. Jacobi's Identity, 224

10.10. Lagrange's Brackets, 225

10.11. Liouville's Theorem, 226

Summary, 227

Worked Out Examples, 229

Exercises. 231

Chapter-I

Vector

1.1. Fundamental Concept of Scalar and Vectors:

Any physical quantity· having both direction and magnitude is called vector

quantity and these quantity must obey some fundamental laws of addition and

subtraction.

But the quantity having only magnitude, but no direction is called scalar

quantity. Basically, the magnitude or measure of a scalar quantity is quite

independent of any co-ordinate system, but the measure of vector quantity

depends on the frame choosen.

All scalar quantities are, however, subject to the ordinary algebraical laws

of addition and multiplication:

viz. (i) a + b = b + a

a x b = b x a (Commutative law)

(ii) (a + b) + c = a + (b + c)

(a x b) x c = a x (b xc) (Associative law)

(iii) a x (b + c) = a x b + a x c

(a + b) x c = a Xc + b x c (Distributive law)

On the other hand vectors are subject to the triangular or parallelogram law

of addition (or subtraction).

1.2. Unit Vectors and General representation of a

vector:

A vector is analytically represented by a letter

with an arrow over it, i.e, A and its magnitude is

p

o * Basically, Physical quantities are Tensor. Any tensor is that quantity which may

change under co-ordinate transformatIOn. The tensor of rank 'n' is that quantity

which transforms through 'n' no. of cu-efficient matrices. Actually a vector is a

tensor of rank 1 where as a scalar is a tensor of rank O.