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| ISBN-13: | 9780486140780 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 12/20/2012 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 416 |
| Sales rank: | 735,370 |
| File size: | 17 MB |
| Note: | This product may take a few minutes to download. |
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CLASSICAL MECHANICS
By H.C. CORBEN, PHILIP STEHLE
Dover Publications, Inc.
Copyright © 1977 H. C. Corben and Philip StehleAll rights reserved.
ISBN: 978-0-486-14078-0
CHAPTER 1
KINEMATICS OF PARTICLES
1. INTRODUCTION
The sciences of physics and chemistry are concerned with the interactions between inanimate material systems. Although chemistry is related chiefly to the interactions between atoms to form molecules, physics is primarily the study of the motion induced in one system of matter by the presence of other systems and of the characterization and measurement of such states of motion by a human observer.
The situations of interest to the physicist are among the least complicated of all known phenomena. This fundamental simplicity of the subject matter allows the science to reach and even demand a degree of precision in both theory and experiment not known elsewhere. Sometimes this precision can be realized, however, only by the introduction of complexities of theory and experiment which mask the very simplicity that made it possible.
The aim of theoretical physics is to provide and develop a self-consistent mathematical structure which runs so closely parallel to the development of physical phenomena that, starting from a minimum number of hypotheses, it may be used to accurately describe and even predict the results of all carefully controlled experiments. The desire for accuracy, however, must be tempered by the need for reasonable simplicity, and the theoretical description of a physical situation is always simplified for convenience of analytical treatment. Such simplifications may be thought of as arising both from physical approximations, i.e., the neglect of certain physical effects which are judged to be of negligible importance, and from mathematical approximations made during the development of the analysis. However, these two types of approximations are not really distinct, for usually each may be discussed in the language of the other. Representing as they do an economy rather than an ignorance, such approximations may be refined by a series of increasingly accurate calculations, performed either algebraically or with an analog or digital computer.
More subtle approximations appear in the laws of motion which are assumed as a starting point in any theoretical analysis of a problem. At present the most refined form of theoretical physics is called quantum field theory, and the theory most accurately confirmed by experiment is a special case of quantum field theory called quantum electrodynamics. According to this discipline, the interactions between electrons, positrons, and electromagnetic radiation have been computed and shown to agree with the results of experiment with an over-all accuracy of one part in 109. Unfortunately, analogous attempts to describe the interactions between mesons, hyperons, and nucleons are at present unsuccessful.
These recent developments are built on a solid structure which has been developed over the last three centuries and which is now called classical mechanics. Figures 1-1 and 1-2 illustrate how classical mechanics is related to other basic physical theories. A line connecting two boxes indicates that the branch of theoretical physics to which the arrow points is a particular case of the branch from which the line originates. The solid lines indicate special cases from a mathematical point of view; i.e., they lead from a theory to one of its applications. The dashed lines, on the other hand, refer to particular cases that may be obtained by neglecting one or the other of the three basic effects that have been discovered since the development of nonrelativistic classical theory-radiative effects, quantum effects, and relativistic effects. For some situations (e.g., quantum effects in atomic theory) one of these may be of dominant importance, whereas for others (e.g., celestial mechanics) the same effect may be completely negligible.
A theory that describes the motion of a particle at any level of approximation must eventually reduce to nonrelativistic classical mechanics when conditions are such that relativistic, quantum, and radiative corrections can be neglected. This fact makes the subject basic to the student's understanding of the rest of physics, in the same way that over the centuries it has been the foundation of human, understanding of the behavior of physical phenomena.
Classical mechanics accurately describes the motion of a material system provided that the angular momentum of the system with respect to the nearest system which is influencing its motion is large compared with the quantum unit of angular momentum h = 1.054 × 10-27 gram-cm2/sec. Some examples are given in Table 1-1. Clearly, in all but the last case the existence of a smallest unit of angular momentum is irrelevant, and the error introduced by using the approximation of classical mechanics will be small compared with both unavoidable experimental errors and other errors and approximations made in describing theoretically the actual physical situation. However, classical mechanics should not be studied only as an introduction to the more refined theories, for despite advances made during this century it continues to be the mechanics used to describe the motion of directly observable macroscopic systems. Although an old subject, the mechanics of particles and rigid bodies is finding new applications in a number of areas, including the fields of vacuum and gaseous electronics, accelerator design, space technology, plasma physics, and magnetohydrodynamics. Indeed, more effort is being put into the development of the consequences of classical mechanics today than at any time since it was the only theory known.
Throughout all but the last chapters of this book we are concerned with nonrelativistic classical mechanics. This theory is applicable to the motion of systems which not only possess an angular momentum large compared with [??] but are moving with relative velocities small compared with that of light (c = 2.99776 × 1010 cm/sec). In Chap-ter 16, however, we give an introduction to the special theory of relativity, which is capable of describing situations in which this restriction on the velocity does not necessarily hold.
In addition, none of the above theories is applicable to the description of the interaction between systems separated by distances comparable to Gm/c2, where m is the largest mass and G is the gravitational constant (G = 6.670 × 10-8 dyne-cm2/gram2). For a proton and an electron this distance is extremely small (~10-52 cm) and many unknown properties of the interaction between these particles already occur at very much larger distances (~10-13 cm). For a planet inter-acting with the sun, the critical distance is 1.47 km, so that although Mercury is much farther from the sun than this, there exists in its motion a very small anomaly that cannot be described even by relativistic classical mechanics but which is adequately explained by the general theory of relativity.
2. DEFINITION AND DESCRIPTION OF PARTICLES
The simplest mechanical system is one which may be represented in the mathematical scheme of mechanics by a point. Such a system is called a particle. How good a representation this is for a particular system can be determined only when we come to consider a more complicated representation of the system and can estimate the effects due to features neglected in the particle treatment. Intuitively it is plausible that it will be a good approximation for a given body provided that this body is small compared with its distance from other bodies and provided that the motion of the body as a whole is only slightly affected by its internal motion. Thus a particle is a body whose internal motion is irrelevant to its motion as a whole, so that it may be represented by a mass point having no extension in space.
A particle is described when its position in space is given and when the values of certain parameters such as mass, electric charge, and magnetic moment are given. By our definition of a particle, these parameters must have constant values because they describe the internal constitution of the particle. If these parameters do vary with time, we are not dealing with a simple particle. The position of a particle may, of course, vary with time.
We may specify the position of a single particle in space by giving its distances from each of three mutually perpendicular planes. These three numbers are called the cartesian coordinates of the particle. Since three mutually perpendicular planes meet in three mutually perpendicular lines, we may also consider the cartesian coordinates of a particle as the displacements in the directions of these three lines needed to move the particle from the point of intersection of the three lines to its actual position. These coordinates will be denoted by x1, x2, x3, or in general by xi (i = 1, 2, 3). To avoid the use of indices in particular problems, we shall also use the letters x, y, z as coordinates.
The coordinate system introduced in this way plays the role of the observer of the particle in the mathematical scheme. The unessential features of the observer have been completely removed, only an extreme simplification of the observer's measuring implements in the form of a set of coordinates being left. The very complex relationship between the particle and the observer which is to be described mathematically is reduced to the relationship of a point to a set of coordinates. In this way we tend almost to forget the observer entirely. In quantum mechanics and relativity theory the role of the observer himself is stressed considerably more.
The coordinates xi may be considered as the components of a vector, the radius vector r of the particle
(2.1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where ei is the unit vector along the i coordinate axis. Sometimes these vectors are denoted by i, j, k respectively.
We now introduce a notation that makes many formulas simpler in appearance and easier to understand. Whenever an italic index appears twice, it is to be summed over the entire range of the index. Greek indices are not to be summed unless a summation is explicitly indicated. We shall usually reserve such indices for numbering individual particles in many-particle systems. In this notation (2.1) becomes simply
(2.2)
r = eixi
If N particles are present, the position of each one is specified by its radius vector, that of the ρth particle being denoted by rρ. The 3N = f' quantities xρ i (i = 1, 2, 3; ρ = 1, 2, ..., N) specify the configuration of the entire system. For many purposes it is more convenient to regard the f ' quantities xρ,i as the cartesian coordinates of a single particle which is located in an f'-dimensional space. (For reasons which will be explained in Sec. 10, coordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are frequently used, mp being the mass of the pth particle.) This f'-dimensional space is called the unconstrained configuration space of the system.
Very often it is not convenient to describe the position of even a single particle in terms of rectangular cartesian coordinates referred to a particular set of coordinate axes. If, for example, the particle moves in a plane under the influence of a force which is directed toward a fixed point in the plane and which is independent of the azimuthal angle θ, it is usually more convenient to use the plane polar coordinates
(2.3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
or if the force is spherically symmetrical it is natural to use the spherical coordinates
(2.4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
These coordinates are also used if the particle is constrained to move on a fixed circle or fixed sphere, respectively.
Sometimes it is useful to look at the motion of the particle from the point of view of a set of coordinates in uniform motion in the x direction with velocity v
(2.5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
or from a uniformly accelerated system
(2.6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In general, each transformation of the coordinate system xi to a new set qm may be expressed as a set of three equations of the form
(2.7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For the stationary coordinate systems (2.3) and (2.4), the relations between xi and qi do not involve the time t.
If equations (2.7) are such that the three coordinates qm can be expressed as functions of the xi, we have
(2.8)
qm = qm(x1, x2, x3, t)
The qm are as effective as the xi in describing the position of the particle. The qm are called generalized coordinates of the particle. The generalized coordinates may themselves be rectangular cartesian coordinates [as in (2.5), (2.6)] or they may be a set of any three variables, not necessarily with the dimension of length, which between them specify unambigously the position of the particle relative to some set of axes.
To obtain the necessary and sufficient conditions that equations (2.7) ay be solved for the qm at a fixed value of time to give a set of equations of the form (2.8), we consider a small displacement of the particle defined by changes δxi in the xi, with t held fixed:
(2.9)
δxi = [partial derivative]xi/[partial derivative]qm δqm
(This is, according to our notation, the sum of three terms!) Equation (2.9) is a set of inhomogeneous linear equations for the δqm which can be solved if and only if the determinant of the coefficients does not vanish:
(2.10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(Continues...)
Excerpted from CLASSICAL MECHANICS by H.C. CORBEN, PHILIP STEHLE. Copyright © 1977 H. C. Corben and Philip Stehle. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Chapter 1. Kinematics of Particles1. Introduction
2. Definition and Description of Particles
3. Velocity
4. Acceleration
5. Special Coordinate Systems
6. Vector Algebra
7. Kinematics and Measurement
Exercises
Chapter 2. The Laws of Motion
8. Mass
9. Momentum and Force
10. Kinetic Energy
11. Potential Energy
12. Conservation of Energy
13. Angular Momentum
14. Rigid Body Rotating about a Fixed Point
15. A Theorem on Quadratic Functions
16. Inertial and Gravitational Masses
Exercises
Chapter 3. Conservative Systems with One Degree of Freedom
17. The Oscillator
18. The Plan Pendulum
19. Child-Langmuir Law
Exercises
Chapter 4. Two-Particle Systems
20. Introduction
21. Reduced Mass
22. Relative Kinetic Energy
23. Laboratory and Center-of-Mass Systems
24. Central Motion
Exercises
Chapter 5. Time-Dependent Forces and Nonconservative Motion
25. Introduction
26. The Inverted Pedulum
27. Rocket Motion
28. Atmospheric Drag
29. The Poynting-Robertson Effect
30. The Damped Oscillator
Exercises
Chapter 6. Lagrange's Equations of Motion
31. Derivation of Lagrange's Equations
32. The Lagrangian Function
33. The Jacobian Integral
34. Momentum Integrals
35. Charged Particle in an Electromagnetic Field
Exercises
Chapter 7. Applications of Lagrange's Equations
36. Orbits under a Central Force
37. Kepler Motion
38. Rutherford Scattering
39. The Spherical Pendulum
40. Larmor's Theorem
41. The Cylindrical Magnetron
Exercises
Chapter 8. Small Oscillations
42. Oscillations of a Natural System
43. Systems with Few Degrees of Freedom
44. "The Stretched String, Discrete Masses"
45. Reduction of the Number of Degrees of Freedom
46. Laplace Transforms and Dissipative Systems
Exercises
Chapter 9. Rigid Bodies
47. Displacements of a Rigid Body
48. Euler's Angles
49. Kinematics of Rotation
50. The Momental Ellipsoid
51. The Free Rotator
52. Euler's Equations of Motion
Exercises
Chapter 10. Hamiltonian Theory
53. Hamilton's Equations
54. Hamilton's Equations in Various Coordinate Systems
55. Charged Particle in an Electromagnetic Field
56. The Virial Theorem
57. Variational Principles
58. Contact Transformations
59. Alternative Forms of Contact Transformations
60. Alternative Forms of the Equations of Motion
Exercises
Chapter 11. The Hamilton-Jacobi Method
61. The Hamilton-Jacobi Equation
62. Action and Angle Variables-Periodic Systems
63. Separable Mulitply-Periodic Systems
64. Applications
Exercises
Chapter 12. Infinitesimal Contact Transformations
65. Transformation Theory of Classical Dynamics
66. Poisson Brackets
67. Jacobi's Identity
68. Poisson Brackets in Quantum Mechanics
Exercises
Chapter 13. Further Development of Transformation Theory
69. Notation
70. Integral Invariants and Liouville's Theorem
71. Lagrange Brackets
72. Change of Independent Variable
73. Extended Contact Transformations
74. Perturbation Theroy
75. Stationary State Perturbation Theory
76. Time-Dependent Perturbation Theory
77. Quasi Coordinates and Quasi Momenta
Exercises
Chapter 14. Special Applications
78. Noncentral Forces
79. Spin Motion
80. Variational Principles in Rocket Motion
81. The Boltzmann and Navier-Stokes Equations
Chapter 15. Continuous Media and Fields
82. The Stretched String
83. Energy-Momentum Relations
84. Three-Dimensional Media and Fields
85. Hamiltonian Form of Field Theory
Exercises
Chapter 16. Introduction to Special Relativity Theory
86. Introduction
87. Space-Time and Lorentz Transformation
88. The Motion of a Free Particle
89. Charged Particle in an Electromagnetic Field
90. Hamiltonian Formulation of the Equations of Motion
91. Transformation Theory and the Lorentz Group
92. Thomas Precession
Exercises
Chapter 17. The Orbits of Particles in High Energy Accelerators
93. Introduction
94. Equilibrium Orbits
95. Betatron Oscillations
96. Weak Focusing Accelerators
97. Strong Focusing Accelerators
98. Acceleration and Synchrotron Oscillations
Appendix I Riemannian Geometry
Appendix II Linear Vector Spaces
Appendix III Group Theory and Molecular Vibrations
Apendix IV Quaternions and Pauli Spin Matrices
Index