G. Aruldhas [Aruldhas - Classical Mechanics

Clas s ical Mechanics

G. Aruldhas

Formerly

Professor & Head of Physics & Dean Faculty of Science

University of Kerala

New Delhi-110001

2008

CLASSICAL MECHANICS

G. Aruldhas

2008 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher.

ISBN-978-81-203-3331-4

The export rights of this book are vested solely with the publisher.

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Jay Print Pack Private Limited, New Delhi-110015.

To

Myrtle and our children

Vinod & Anitha, Manoj & Bini, Ann & Suresh

Contents

Preface xi

1.1 Frames of Reference

Cartesian Co-ordinates ( x , y , z )

Plane Polar Co-ordinates ( r , q )

Cylindrical Co-ordinates ( r , f , z )

Spherical Polar Co-ordinates ( r , q , f )

1.2 Newtons Laws of Motion

Newtons First Law of Motion

Newtons Second Law of Motion

Newtons Third Law of Motion

1.3 Inertial and Non-inertial Frames

1.4 Mechanics of a Particle

Conservation of Linear Momentum

Angular Momentum and Torque

Conservation of Angular Momentum

Work Done by a Force

Conservative Force

Conservation of Energy

1.5 Motion under a Constant Force

1.6 Motion under a Time-dependent Force

1.7 Reflection of Radiowaves from the Ionosphere

1.8 Motion under a Velocity Dependent Force

1.9 Motion of Charged Particles in Magnetic Fields

Worked Examples

Review Questions

Problems

2.1 Centre of Mass

2.2 Conservation of Linear Momentum

2.3 Angular Momentum

2.4 Conservation of Angular Momentum

2.5 Kinetic Energy for a System of Particles

2.6 Energy Conservation of a System of Particles

2.7 Time Varying Mass SystemsRockets

Worked Examples

Review Questions

Problems

3.1 Constraints

Holonomic Constraints

Non-holonomic Constraints

Scleronomous and Rheonomous Constraints

3.2 Generalized Co-ordinates

Degrees of Freedom

Generalized Co-ordinates

Configuration Space

3.3 Principle of Virtual Work

3.4 DAlemberts Principle

3.5 Lagranges Equations

3.6 Kinetic Energy in Generalized Co-ordinates

3.7 Generalized Momentum

3.8 First Integrals of Motion and Cyclic Co-ordinates

Cyclic Co-ordinates

3.9 Conservation Laws and Symmetry Properties

Homogeneity of Space and Conservation of Linear Momentum

Isotropy of Space and Conservation of Angular Momentum

Homogeneity of Time and Conservation of Energy

3.10 Velocity-dependent Potential

3.11 Dissipative Force

3.12 Newtonian and Lagrangian Formalisms

Worked Examples

Review Questions

Problems

4.1 Hamiltons Principle

4.2 Deduction of Hamiltons Principle

4.3 Lagranges Equation from Hamiltons Principle

4.4 Hamiltons Principle for Non-holonomic Systems

Worked Examples

Review Questions

Problems

5.1 Reduction to One-body Problem

5.2 General Properties of Central Force Motion

Angular Momentum

Law of Equal Areas

5.3 Effective Potential

5.4 Classification of Orbits

5.5 Motion in a Central Force FieldGeneral Solution

Energy Method

Lagrangian Analysis

5.6 Inverse Square Law Force

5.7 Keplers Laws

5.8 Law of Gravitation from Keplers Laws

5.9 Satellite Parameters

5.10 Communication Satellites

5.11 Orbital Transfers

5.12 Scattering in a Central Force Field

5.13 Scattering Problem in Laboratory Co-ordinates

Worked Examples

Review Questions

Problems

6.1 The Hamiltonian of a System

6.2 Hamiltons Equations of Motion

6.3 Hamiltons Equations from Variational Principle

6.4 Integrals of Hamiltons Equations

Energy Integral

Integrals Associated with Cyclic Co-ordinates

6.5 Canonical Transformations

6.6 Poisson Brackets

Fundamental Poisson Brackets

Fundamental Properties of Poisson Brackets

Equations of Motion in Poisson Bracket Form

6.7 Poisson Bracket and Integrals of Motion

6.8 The Canonical Invariance of Poisson Bracket

6.9 Lagrange Brackets

6.10 D -Variation

6.11 The Principle of Least Action

Different Forms of Least Action Principle

6.12 Poisson Brackets and Quantum Mechanics

Worked Examples

Review Questions

Problems

7.1 HamiltonJacobi Equation

Physical Significance of S

7.2 Hamiltons Characteristic Function

7.3 Harmonic Oscillator in The H- j Method

7.4 Separation of Variables in The H- j Equation

7.5 Central Force Problem in Plane Polar Co-ordinates

7.6 Action-Angle Variables

7.7 Harmonic Oscillator in Action-Angle Variables

7.8 Kepler Problem in Action-Angle Variables

7.9 Road to Quantization

Worked Examples

Review Questions

Problems

8.1 Introduction

8.2 Angular Momentum

8.3 Kinetic Energy

8.4 Inertia Tensor

8.5 Principal Axes

8.6 Eulers Angles

8.7 Infinitesimal Rotations

8.8 Rate of Change of a Vector

8.9 Coriolis Force

8.10 Eulers Equations of Motion

8.11 Force-free Motion of a Symmetrical Top

8.12 Heavy Symmetric Top with One Point Fixed

Worked Examples

Review Questions

Problems

9.1 Equilibrium and Potential Energy

9.2 Theory of Small Oscillations

9.3 Normal Modes

9.4 Two Coupled Pendula

Resonant Frequencies

Normal Modes

9.5 Longitudinal Vibrations of CO Molecule

Normal Frequencies

Normal Modes

Normal Co-ordinates

Worked Examples

Review Questions

Problems

10.1 Galilean Transformation

10.2 Electromagnetism and Galilean Transformation

10.3 MichelsonMorley Experiment

The Interferometer

The Experiment

10.4 The Postulates of Special Theory of Relativity

10.5 Lorentz Transformation

10.6 Velocity Transformation

10.7 Length Contraction

10.8 Time Dilation

10.9 Simultaneity

10.10 Mass in Relativity

10.11 Mass and Energy

10.12 Relativistic Lagrangian of a Particle

10.13 Relativistic Hamiltonian of a Particle

10.14 Space-Time Diagram

10.15 Geometrical Interpretation of Lorentz Transformation

10.16 Principle of Covariance

10.17 Four-Vectors in Mechanics

Position Four-Vector

Four-Velocity

Momentum Four-Vector

Four-Force

Four-Acceleration

10.18 Charge Current Four-Vector

10.19 Invariance of Maxwells Equations

Maxwells Equations

Vector and Scalar Potentials

Gauge Transformations

Four-Vector Potential

10.20 Electromagnetic Field Tensor

10.21 General Theory of Relativity

Principle of Equivalence

Bending of Light in a Gravitational Field

Precession of the Perihelion of Planetary Orbits

Space Curvature

Gravitational Red Shift

Worked Examples

Review Questions

Problems

11.1 Linear and Nonlinear Systems

11.2 Integration of Linear Equation: Quadrature Method

11.3 Integration of Nonlinear Second Order Equation

11.4 The Pendulum Equation

11.5 Phase Plane Analysis of Dynamical Systems

Phase Curve of Simple Harmonic Oscillator

Phase Curve of Damped Oscillator

11.6 Phase Portrait of the Pendulum

11.7 Matching of Phase Curve with Potential V ( x )