Elements of Newtonian Mechanics: Including Nonlinear Dynamics

ISBN-13: 9783540608417 / Angielski / Miękka / 1996 / 437 str.

J. M. Knudsen; Jens M. Knudsen; Poul G. Hjorth

Opis

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. The book still contains a one-semester (15 weeks) first university course on Newtonian mechanics. This necessarily introduces some constraints on the choice of topics and the level of mathematical sophistication expected from the reader. If one looks for discussions of technical issues, such as the physics behind various manifestations of friction, or the tensorial nature of the rotation vector, one will look in vain. The book contains what we feel are the essential aspects of Newtonian Mechanics. It is a pleasure again to thank Springer-Verlag and in particular Dr. H. J. KOisch and the staff at the Heidelberg office for helpfulness and professional collaboration.

Spis treści

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 Newton’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.- 11.3.1 The Parallel Axis Theorem.- 11.3.2 The Perpendicular Axis Theorem.- 11.4 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.- 11.5.5 Rectangular Plate.- 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.- 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 Kepler’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.- 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 Nonlinear 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.

Blurb

This textbook provides a thorough introduction to Newtonian Mechanics and is intended for university students in physics, astronomy and engineering. It is based on a course for which Dr Knudsen earned an award as the best introductory course at the university of Copenhagen, Denmark. About 190 problems with solutions and 84 worked examples help the student to grasp the essential aspects of the subject, and to develop proficiency in using the powerful methods of Newtonian mechanics. Moreover, the problem of motion and the concept of inertial frames is faced from the very beginning, and runs through the entire development of classical mechanics. This new and fresh approach is presented in its second edition, which has been revised and enlarged by a new chapter on nonlinear dynamics.