## Classical MechanicsThis is the fifth edition of a well-established textbook. It is intended to provide a thorough coverage of the fundamental principles and techniques of classical mechanics, an old subject that is at the base of all of physics, but in which there has also in recent years been rapid development. The book is aimed at undergraduate students of physics and applied mathematics. It emphasizes the basic principles, and aims to progress rapidly to the point of being able to handle physically and mathematically interesting problems, without getting bogged down in excessive formalism. Lagrangian methods are introduced at a relatively early stage, to get students to appreciate their use in simple contexts. Later chapters use Lagrangian and Hamiltonian methods extensively, but in a way that aims to be accessible to undergraduates, while including modern developments at the appropriate level of detail. The subject has been developed considerably recently while retaining a truly central role for all students of physics and applied mathematics.This edition retains all the main features of the fourth edition, including the two chapters on geometry of dynamical systems and on order and chaos, and the new appendices on conics and on dynamical systems near a critical point. The material has been somewhat expanded, in particular to contrast continuous and discrete behaviours. A further appendix has been added on routes to chaos (period-doubling) and related discrete maps. The new edition has also been revised to give more emphasis to specific examples worked out in detail.Classical Mechanics is written for undergraduate students of physics or applied mathematics. It assumes some basic prior knowledge of the fundamental concepts and reasonable familiarity with elementary differential and integral calculus. |

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### Contents

Introduction | 1 |

11 Space and Time | 2 |

12 Newtons Laws | 5 |

13 The Concepts of Mass and Force | 10 |

14 External Forces | 13 |

Linear Motion | 17 |

22 Motion near Equilibrium the Harmonic Oscillator | 20 |

23 Complex Representation | 24 |

97 Instantaneous Angular Velocity | 216 |

98 Rotation about a Principal Axis | 218 |

99 Eulers Angles | 221 |

910 Summary | 225 |

Lagrangian Mechanics | 231 |

102 Lagranges Equations | 233 |

103 Precession of a Symmetric Top | 236 |

104 Pendulum Constrained to Rotate about an Axis | 238 |

24 The Law of Conservation of Energy | 25 |

25 The Damped Oscillator | 27 |

26 Oscillator under Simple Periodic Force | 30 |

27 General Periodic Force | 34 |

28 Impulsive Forces the Greens Function Method | 37 |

29 Collision Problems | 39 |

210 Summary | 42 |

Energy and Angular Momentum | 49 |

32 Projectiles | 51 |

33 Moments Angular Momentum | 53 |

34 Central Forces Conservation of Angular Momentum | 55 |

35 Polar Coordinates | 57 |

36 The Calculus of Variations | 59 |

37 Hamiltons Principle Lagranges Equations | 62 |

38 Summary | 66 |

Central Conservative Forces | 73 |

42 The Conservation Laws | 76 |

43 The Inverse Square Law | 78 |

44 Orbits | 84 |

45 Scattering Crosssections | 90 |

46 Mean Free Path | 94 |

47 Rutherford Scattering | 96 |

48 Summary | 98 |

Rotating Frames | 105 |

52 Particle in a Uniform Magnetic Field | 108 |

53 Acceleration Apparent Gravity | 111 |

54 Coriolis Force | 114 |

55 Larmor Effect | 120 |

56 Angular Momentum and the Larmor Effect | 121 |

57 Summary | 124 |

Potential Theory | 129 |

62 The Dipole and Quadrupole | 131 |

63 Spherical Charge Distributions | 134 |

64 Expansion of Potential at Large Distances | 137 |

65 The Shape of the Earth | 140 |

66 The Tides | 144 |

67 The Field Equations | 148 |

68 Summary | 152 |

The TwoBody Problem | 159 |

72 The Centreofmass Frame | 162 |

73 Elastic Collisions | 165 |

74 CM and Lab Crosssections | 168 |

75 Summary | 173 |

ManyBody Systems | 177 |

82 Angular Momentum Central Internal Forces | 181 |

83 The EarthMoon System | 183 |

84 Energy Conservative Forces | 188 |

85 Lagranges Equations | 190 |

86 Summary | 192 |

Rigid Bodies | 197 |

92 Rotation about an Axis | 198 |

93 Perpendicular Components of Angular Momentum | 203 |

94 Principal Axes of Inertia | 205 |

95 Calculation of Moments of Inertia | 208 |

96 Effect of a Small Force on the Axis | 211 |

105 Charged Particle in an Electromagnetic Field | 241 |

106 The Stretched String | 244 |

107 Summary | 248 |

Small Oscillations and Normal Modes | 253 |

112 Equations of Motion for Small Oscillations | 256 |

113 Normal Modes | 258 |

114 Coupled Oscillators | 261 |

115 Oscillations of Particles on a String | 266 |

116 Normal Modes of a Stretched String | 269 |

117 Summary | 272 |

Hamiltonian Mechanics | 277 |

122 Conservation of Energy | 280 |

123 Ignorable Coordinates | 282 |

124 General Motion of the Symmetric Top | 285 |

125 Liouvilles Theorem | 289 |

126 Symmetries and Conservation Laws | 291 |

127 Galilean Transformations | 295 |

128 Summary | 300 |

Dynamical Systems and Their Geometry | 307 |

132 Firstorder Systems the Phase Line n 1 | 309 |

133 Secondorder Systems the Phase Plane n 2 | 312 |

134 PreyPredator Competingspecies Systems and War | 318 |

135 Limit Cycles | 324 |

136 Systems of Third and Higher Order | 329 |

137 Sensitivity to Initial Conditions and Predictability | 337 |

138 Summary | 340 |

Order and Chaos in Hamiltonian Systems | 347 |

142 Surfaces of Section | 351 |

143 ActionAngle Variables | 354 |

144 Some Hamiltonian Systems which Exhibit Chaos | 359 |

145 Slow Change of Parameters Adiabatic Invariance | 369 |

146 Nearintegrable Systems | 372 |

147 Summary | 374 |

Vectors | 381 |

A2 The Scalar Product | 384 |

A3 The Vector Product | 385 |

A4 Differentiation and Integration of Vectors | 388 |

A5 Gradient Divergence and Curl | 390 |

A6 Integral Theorems | 393 |

A7 Electromagnetic Potentials | 397 |

A8 Curvilinear Coordinates | 398 |

A9 Tensors | 401 |

A10 Eigenvalues Diagonalization of a Symmetric Tensor | 403 |

Conics | 409 |

B2 Polar Form | 412 |

Phase Plane Analysis near Critical Points | 415 |

C2 Almost Linear Systems | 421 |

C3 Systems of Third and Higher Order | 423 |

Discrete Dynamical Systems Maps | 425 |

D2 Twodimensional Maps | 433 |

D3 Twist Maps and Torus Breakdown | 437 |

Answers to Problems | 445 |

463 | |

465 | |

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### Common terms and phrases

2ero acceleration amplitude angle angular momentum angular velocity asymptotically stable attractor axes axis centre of mass Chapter collision components conservation laws conservative force constant Coriolis force corresponding critical point cross-section curve define degrees of freedom differential direction discussion distance dynamical Earth effect eigenvalues ellipse equal equations of motion equilibrium example external forces frame frequency generali2ed co-ordinates given gravitational Hamilton's equations Hamiltonian harmonic oscillator Hence hori2ontal inertia initial conditions integral kinetic energy Lagrange's equations Lagrangian function linear magnetic field maxiumm miniumm momenta Moon normal modes Note obtain orbit origin parameters particle moving particle of mass period phase plane phase portrait phase space polar position potential energy potential energy function precession Problem radial radius rate of change relative rigid body rotating scalar scattering Show small oscillations solution sphere surface symmetry tensor theorem trajectories umch umltiplying umst unstable vanish variables vector vertical write