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# Elements of Newtonian Mechanics: Including Nonlinear Dynamics

## byJens M. Knudsen, Poul G. Hjorth

### Paperback | January 17, 2001

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In the third edition a number of minor misprints that appeared in the second edition have have been corrected. Furthermore, 17 new problems have been added, at the end of chapters 6, 8, 9, 11, 12, 13, and 14. The answers to these 17 problems have not been listed in the 'Answers' section at the end of the book. This will permit the problems to be used as hand-in problems or perhaps in mid-term exams. JMK '¬9 PGH Copenhagen May 2000 Preface to the Second Edition In the second edition, a number of misprints that appeared in the first edition have been corrected. In addition to this, we have made improvements based on the experience gathered in the use of the first English edition of the book in the introductory course in physics at the University of Copenhagen. A chapter introducing nonlinear dynamics has been added. The purpose of this chapter is to provide supplementary reading for the students who are interested in this area of active research, where Newtonian mechanics plays an essential role. The students who wish to dig deeper, should consult texts dedicated to the study of nonlinear dynamical systems and chaos. The literature list at the end of this book contains several references for the topic.

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Title:Elements of Newtonian Mechanics: Including Nonlinear DynamicsFormat:PaperbackDimensions:452 pagesPublished:January 17, 2001Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:354067652X

ISBN - 13:9783540676522

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Table of Contents

1. The Foundation of Classical Mechanics.- 1.1 Principia.- 1.2 Prerequisites for Newton.- 1.3 The Masterpiece.- The Acceleration of Gravity.- Circular Motion.- Communication Satellite.- Horizontal Throw.- The Gravitational Constant.- String Force.- Forces and Tension.- Dimensional Analysis.- 1.4 Concluding Remarks.- 1.5 Problems.- 2. Newton's Five Laws.- 2.1 Newton's Laws of Motion.- 2.2 Integration of the Equation of Motion.- Constant Force.- The Harmonic Oscillator.- Mass on a Spring in the Gravitational Field of Earth.- Sphere Falling Through a Liquid.- Solid Against Solid.- The Atwood Machine.- Force in Harmonic Motion.- Charged Particle in a Uniform Magnetic Field.- Thomson's Experiment.- Work and Energy in Linear Motion of a Particle.- Free Fall Towards the Sun from a Great Distance.- Momentum Conservation.- Inelastic Collisions.- Rocket Propulsion.- Some Qualitative Remarks on Rocket Propulsion.- Ball Against a Wall.- 2.3 Problems.- 3. Gravitational and Inertial Mass.- 3.1 Gravitational Mass.- 3.2 Inertial Mass.- 3.3 Proportionality Between Inertial and Gravitational Mass.- 3.4 Newtonl's Experiment.- The Satellite.- An Elevator in Free Fall.- Three Balls.- 3.5 Problem.- 4. The Galilei Transformation.- 4.1 The Galilei Transformation.- 4.2 Galileo Speaks.- Velocity Transformation.- 4.3 Problems.- 5. The Motion of the Earth.- 5.1 Examples.- Vectors and the Rotation of a Rigid Body.- Angular Velocities in the Solar System.- 5.2 Problems.- 6. Motion in Accelerated Reference Frames.- 6.1 Newton's 2nd Law Within Accelerated Reference Frames.- 6.2 The Equivalence Principle of Mechanics.- 6.3 The Einstein Box.- Balloon in Accelerated Frame.- Mass on an Oscillating Plate.- Pendulum in an Elevator.- 6.4 The Centrifugal Force.- Earth's Orbit Around the Sun.- Grass on a Rotating Disk.- The Variation of g with Latitude.- 6.5 Tidal Fields.- The Roche Limit.- 6.6 The Coriolis Force.- Coriolis Force on a Train.- Particle on a Frictionless Disc.- The Vertical Throw.- 6.7 Tidal Forces and Local Inertial Frames.- Global and Local Inertial Frames.- 6.8 The Foucault Pendulum.- 6.9 Newton's Bucket.- 6.10 Review: Fictitious Forces.- 6.11 Problems.- 7. The Problem of Motion.- 7.1 Kinematic and Dynamic Views of the Problem of Motion.- 7.2 Einstein Speaks.- 7.3 Symmetry.- 7.4 The Symmetry (Invariance) of Newton's 2nd Law.- 7.5 Limited Absolute Space.- 7.6 The Asymmetry (Variance) of Newton's 2nd Law.- 7.7 Critique of the Newtonian View.- 7.8 Concluding Remarks.- 8. Energy.- 8.1 Work and Kinetic Energy.- 8.2 Conservative Force Fields.- 8.3 Central Force Fields.- 8.4 Potential Energy and Conservation of Energy.- 8.5 Calculation of Potential Energy.- Constant Gravitational Field.- Spring Force.- Gravity Outside a Homogeneous Sphere.- 8.6 The Gravitational Field Around a Homogeneous Sphere.- 8.6.1 The Field Around a Spherical Shell.- 8.6.2 A Solid Sphere.- 8.7 Examples.- Particle on a Frictionless Curve.- String Force in the Pendulum.- The Gravitational Potential Outside the Earth.- Potential Energy Due to Electric Forces.- A Tunnel Through the Earth.- The Asymmetry of Nature.- 8.8 Review: Conservative Forces and Potential Energy.- 8.9 Problems.- 9. The Center-of-Mass Theorem.- 9.1 The Center of Mass.- 9.2 The Center-of-Mass Frame.- 9.3 Examples.- Two Masses Connected with a Spring.- Inelastic Collisions.- The Collision Approximation.- Freely Falling Spring.- The Wedge.- 9.4 Review: Center of Mass and Center-of-Mass Theorems.- 9.5 Comments on the Conservation Theorems.- 9.6 Problems.- 10. The Angular Momentum Theorem.- 10.1 The Angular Momentum Theorem for a Particle.- 10.2 Conservation of Angular Momentum.- 10.3 Torque and Angular Momentum Around an Axis.- 10.4 The Angular Momentum Theorem for a System of Particles.- 10.5 Center of Gravity.- 10.6 Angular Momentum Around the Center of Mass.- 10.7 Review: Equations of Motion for a System of Particles.- 10.8 Examples of Conservation of Angular Momentum.- Particle in Circular Motion.- Rotation of Galaxies, Solar Systems, etc.- 11. Rotation of a Rigid Body.- 11.1 Equations of Motion.- 11.2 The Rotation Vector.- 11.3 Kinetic Energy of a Rotating Disk.- 11.3.1 The Parallel Axis Theorem.- 11.3.2 The Perpendicular Axis Theorem.- 11.4 Angular Momentum of an Arbitrary Rigid Body in Rotation Around a Fixed Axis.- 11.4.1 The Parallel Axis Theorem in General Form.- 11.5 Calculation of the Moment of Inertia for Simple Bodies.- 11.5.1 Homogenous Thin Rod.- 11.5.2 Circular Disk.- 11.5.3 Thin Spherical Shell.- 11.5.4 Homogenous (Solid) Sphere, Mass M and Radius R 248.- 11.5.5 Rectangular Plate 249.- 11.6 Equation of Motion for a Rigid Body Rotating Around a Fixed Axis.- 11.6.1 Conservation of Angular Momentum.- 11.7 Work and Power in the Rotation of a Rigid Body Around a Fixed Axis.- 11.7.1 Torsion Pendulum.- 11.8 The Angular Momentum Theorem Referred to Various Points.- 11.9 Examples.- Rotating Cylinder.- Falling Cylinder.- The Atwood Machine.- The Physical Pendulum.- The Rod.- 11.10 Review: Linear Motion and Rotation About a Fixed Axis. 266 11.11 Problems.- 12. The Laws of Motion.- 12.1 Review: Classical Mechanics.- 12.2 Remarks on the Three Conservation Theorems.- 12.3 Examples.- Conservation of Angular Momentum.- Rotating Rod.- Man on Disk.- The Sprinkler.- Rolling.- Yo- Yo on the Floor.- Rolling Over an Edge.- Determinism and Predictability.- 12.4 Problems.- 13. The General Motion of a Rigid Body.- 13.1 Inertia in Rotational Motion.- The Dumbbell.- Flywheel on an Axis.- Precession of a Gyroscope.- 13.2 The Inertia Tensor.- The Dumbbell Revisited.- 13.3 Euler's Equations.- 13.3.1 Derivation of Euler's Equations.- 13.4 Kinetic Energy.- 13.5 Determination of the Principal Coordinate System.- Rotating Dumbbell.- Flywheel.- The Gyroscope.- Gyroscope Supported at the Center of Mass.- The Earth as a Gyroscope.- 13.6 Problems.- 14. The Motion of the Planets.- 14.1 Tycho Brahe.- 14.2 Kepler and the Orbit of Mars.- 14.2.1 The Length of a Martian Year.- 14.2.2 The Orbit of the Planet Mars.- 14.2.3 Determination of Absolute Distance in the Solar System.- 14.3 Conic Sections.- 14.4 Newton's Law of Gravity Derived from Kepler's Laws.- 14.5 The Kepler Problem.- 14.5.1 Derivation of Kepler's 3rd Law from Newton's Law of Gravity.- 14.6 The Effective Potential.- 14.7 The Two-Body Problem.- 14.7.1 The Two-Body Problem and Kepler2019;s 3rd Law.- 14.8 Double Stars: The Motion of the Heliocentric Reference Frame.- 14.9 Review: Kepler Motion.- 14.10 Examples.- Planetary Orbits and Initial Conditions.- Shape and Size of Planetary Orbits.- Motion Near the Surface of the Earth.- Velocities in an Elliptical Orbit.- Hohman Orbit to Mars.- The Face of the Moon (Spin-Orbit Locking).- 14.11 Problems.- 15. Harmonic Oscillators.- 15.1 Small Oscillations.- 15.2 Energy in Harmonic Oscillators.- 15.3 Free Damped Oscillations.- 15.3.1 Weakly Damped Oscillations.- 15.3.2 Strongly Damped Oscillations.- 15.3.3 Critical Damping.- 15.4 Energy in Free, Weakly Damped Oscillations.- 15.5 Forced Oscillations.- 15.6 The Forced Damped Harmonic Oscillator.- 15.7 Frequency Characteristics.- 15.7.1 ? ? ?0: A Low Driving Frequency.- 15.7.2 ? ? ?0: A High Driving Frequency.- 15.7.3 ? ? ?0: Resonance.- 15.8 Power Absorption.- 15.9 The Q-Value of a Weakly Damped Harmonic Oscillator.- 15.10 The Lorentz Curve.- 15.11 Complex Numbers.- 15.12 Problems.- 16. Remarks on Nonlinearity and Chaos.- 16.1 Determinism vs Predictability.- 16.2 Linear and Nonliner Differential Equations.- Superposition.- 16.3 Phase Space.- The Simple Harmonic Oscillator.- Phase Space of the Pendulum.- Bifurcation in a Nonlinear Model.- 16.4 A Forced, Damped Nonlinear Oscillator.- 16.5 Liapunov Exponents.- 16.6 Chaos in the Solar System.- 16.7 Problems.- Appendix. Vectors and Vector Calculus.- Selected References.- Answers to Problems.

Editorial Reviews

This book provides an excellent introduction to Newtonian mechanics at an undergraduate level, it is used at University of Copenhague. All the classical topics are presented in a very clear way an discussed with many physicaI examples. The book is divided in 16 chapters and contains more than 100 problerns, of various difficulties, with solutions. It's pedagogical conception is absolutely remarkable, all concepts are carefully introduced and physically discussed with mamy details. Foundation of classical mechanics is first introduced and commented on many problems like acceleration of gravity, circular motion or conmunication satellite. Next chapter is devoted to Newton's laws of motion, more than 30 pages of the book contains examples, on direct applications of these law, from: various fields of physics. Gravitational and inertial mass are discussed in the next chapter and is followed by the concept of Galilei transformation. The next three chapters examine the problem of motion respectively of the Earth, in accelerated reference frames and at the kinematic and dynamic point of view. Energy, Center-of Mass and angular momentum theorem are analysed with some details in chapters 8-10. The problem of rigid body including the motion of the planets and the Kepler Iaws are presented and discussed with great attention. The book ends with the important problem of harmonic oscillators and some remarks on non-linear dynamics. In conclusion I warmly recommend this book to everyone interesting in Newtonian mechanics.S. Metens. Physicalia, 2001/XXXVII/4 "The book presents an excellant description of basic principles of classical mechanics....This book can be useful not only to students, but also to specialists who teach the powerful methods of Newtonian mechanics and the imaginary Newton's world governed by laws of classical mechanics."-Zentralblatt MATH