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Methods of Analytical Dynamics
By Leonard Meirovitch
Dover Publications, Inc.
Copyright © 1998 Leonard MeirovitchAll rights reserved.
ISBN: 9780486137599
CHAPTER 1
Fundamentals of Newtonian Mechanics
The oldest and most fundamental part of physics is concerned with the equilibrium and motion of material bodies. Problems of mechanics attracted the attention of such pioneers of physics as Aristotle and Archimedes. Although the field of classical mechanics is based on only a few fundamental ideas, it has inspired the creation of a large and elegant portion of mathematical analysis. The basic concepts of mechanics are space, time, and mass (or, alternately, force). These concepts are familiar from earlier encounters with the subject of dynamics, but we shall attempt to bring their meaning into sharper focus. To this end, a historical account of the development of mechanics will prove rewarding.
1.1 HISTORICAL SURVEY OF MECHANICS
The development of mechanics can be traced to the development of geometry, kinematics, and dynamics. By far the simplest and most widely used geometry is the geometry of Euclid. A Euclidean space is a space in which the length of a vector is defined by the Pythagorean theorem, according to which the length of the vector squared is equal to the sum of the vector components squared. The parallel and congruence postulates hold, so that Euclidean geometry is ideally suited for the study of the motion of a rigid body. The metrical structure of Euclidean space is homogeneous and isotropic, hence independent of the distribution of matter in the space. As a result, there is complete relativity of position or orientation in that space. A more general geometry, still consistent with the congruence concept, is the geometry of Riemann, which represents an extension of the twodimensional geometry of curved surfaces developed by Gauss. By contrast with Euclidean geometry, in Riemannian geometry the space properties, as reflected by the metric coefficients, can vary from point to point. In such a geometry there is no longer relativity of position. Euclidean geometry was generally accepted as the framework of Newtonian mechanics. By means of a certain transformation, the concepts of Euclidean geometry could be extended to accommodate Einstein's special relativity theory, and it was not until the development of Einstein's general relativity, or gravitational, theory that Euclidean geometry was abandoned in favor of the more general Riemannian geometry.
Kinematics is concerned with the motion of material bodies, and, for this reason, it is sometimes referred to as the geometry of motion. But motion has meaning only when measured relative to a system of reference, which requires a welldefined system of coordinates and a timemeasuring device. In Newtonian mechanics it is assumed that there exists an absolute space which is Euclidean and an absolute time whose flow is independent of the space. Since Euclidean space is homogeneous and isotropic, we must conclude that there is no preferred position or orientation, hence no preferred coordinate system. To measure the time at a certain point, an observer may choose any periodic phenomenon, such as the vibration of a tuning fork. With what is generally referred to as a clock, the question of synchronizing the clock with observers at other points remains. The local time can be extended to other observers by means of signals possessing infinite speed, provided such signals exist.
The study of dynamics considers the motion of material bodies under the influence of the surroundings. The question can be posed whether in the absence of interacting surroundings there exists a natural state of motion for a body. Aristotle explored the idea that there exists a frame of reference such that the natural state for any body is a state of rest with respect to that frame. Indeed, he believed that there was a natural place, namely, the center of the earth, toward which each body was striving and which it approached if no obstacles were encountered. Motion was regarded as starting from rest under the influence of "efficient causes" and determined by "final causes," which implies that once the motion was initiated, the body had a tendency to reach the natural place. Aristotle believed that wherever there was motion there must be a force, with the exception of bodies that moved themselves. This seems to imply that the law of motion he considered can be expressed in terms of modern concepts in the form mass × velocity = force. However, he was still puzzled by the observation that the motion of a body continued even when the source of motion was no longer in contact with the body. Although the notion of a natural place may have had a certain intuitive appeal in ancient Greece, when the earth was believed to be the center of things with the planets moving in circular orbits around it, this idea became questionable when Copernicus showed that the planets were really moving around the sun and not the earth, thus placing the sun at the center. Of course, later it became apparent that the sun did not occupy such a preferred position either, as it could be regarded as but one star among many in the universe. At this point we should recall that the idea of a natural position is in conflict with the concept of a Euclidean space. Aristotle's researches concerning the motion of bodies must be considered a complete failure as no unique frame of reference for velocity is possible.
The first step toward placing the study of dynamics on a truly scientific foundation was taken by Galileo. He set the study of the motion of bodies on the right track by developing the concept of acceleration. Galileo's investigation of falling objects led him to the observation that it is a fallacy to assume that wherever there is motion there must be a force. Indeed he concluded that force causes a change in velocity but no force is necessary to maintain a motion in which the magnitude and direction of the velocity does not change. This is basically the statement of Galileo's law of inertia. The measure of the tendency of a body to resist a change in its uniform motion is known as the mass or inertia of the body. Galileo also observed that while the acceleration is constant, the velocity varies with time for freely falling bodies, so that deviations from a state of rest or uniform motion must be attributed to the influence of other bodies. He recognized that the laws of motion are not affected by uniform motion, so that not only was there no natural position in space but there was no favored velocity of the reference frame either. Thus Galileo showed that there indeed exist preferred reference systems in which a body will move with uniform velocity or remain in a state of rest unless acted upon by external forces. Such a homogeneous and isotropic reference frame is referred to as an inertial space or Galilean reference frame. The inertial system is either at rest or translates with uniform velocity relative to a fixed space. If two systems move uniformly relative to each other, and if one of the systems is inertial, then the other system is also inertial. In fact there is an infinity of inertial frames translating uniformly relative to one another. The properties of space and time are the same and the laws of mechanics are identical in all these frames. This is the essence of Galileo's principle of relativity. It is impossible from observations of mechanical phenomena to detect a uniform motion relative to an inertial space. The relations between the motions expressed in terms of two inertial systems are known as Galilean transformations.
Newton expanded on the ideas of Galileo and toward the end of the seventeenth century formulated what have come to be known as Newton's laws of motion. In fact Newton's first law is simply Galileo's law of inertia. Recognizing that Galileo's results reflected the fact that the gravity force is constant in the vicinity of the earth's surface, Newton generalized these results by admitting variable forces as well. Moreover, Newton applied his laws to the motion of celestial bodies. In the process, he developed his law of gravitation, helped by his correct interpretation of Kepler's planetary laws. Newton's laws of motion assume the simplest form when referred to an inertial system, and, to this end, Newton introduced the notion of absolute space relative to which every motion should be measured. He proposed as an absolute space a coordinate system attached to the distant "fixed stars." Newton's fundamental equations are invariant under Galilean transformations, but this invariance does not hold for transformations involving accelerated reference frames. If we insist on treating mechanical phenomena in accelerated systems, we must introduce fictitious forces, such as centrifugal and Coriolis forces. These fictitious forces are strictly of a kinematical nature and appear when the motion is expressed in terms of rotating coordinate systems. According to Newton, the time is absolute and independent of space, which is another way of saying that the time is the same in any two inertial systems. Moreover, if there are forces exerted by one body upon another which depend only on the relative positions of the bodies, these forces are assumed to propagate instantaneously. Because the time is the same for all inertial frames, the concept of simultaneity presents no difficulty; i.e., if two events are observed to occur simultaneously by one observer traveling with one inertial frame, then the events occur simultaneously for all other inertial observers.
If the principle of relativity is to be valid for all fields of physics, including electrodynamics, then, among other things, the velocity of propagation of light waves, which according to Maxwell's theory of light are special electromagnetic waves, should be the same for all inertial observers. This turns out to be in conflict with the results predicted by observers whose motion with respect to one another are related by means of Galilean transformations. To show this, let us consider two inertial frames translating uniformly with respect to one another and assume that at the origin of one of the systems there is a source emitting light waves in the form of spherical waves centered at the origin and propagating with the speed c. Denoting a unit vector in the radial direction by ur, we can write the velocity of light as seen by an observer traveling with the light source in the form cur. If a second inertial system is moving uniformly with velocity v with respect to the first, and if a Galilean transformation is used, we conclude that an observer on the second system will observe a light velocity cur–v, which is no longer equal in every direction. Hence for that observer the waves do not appear spherical. This, however, is in contradiction with the experimental evidence. Indeed the celebrated experiments performed by Michelson and Morley showed that the velocity of light is the same in all directions and does not depend on the relative motion of the observer and the source. Hence we must conclude that the Galilean transformation cannot be correct and should be replaced by a transformation preserving the constancy of the velocity of light in all systems. Such a transformation, known as the Lorentz transformation, is applicable to both mechanical and electromagnetic phenomena.
By the latter part of the nineteenth century the wave theory of light was set on a firm foundation by the researches of Faraday, Maxwell, and Hertz in electromagnetic field theory. According to this theory, light waves are simply electromagnetic waves propagating with constant velocity relative to an absolute space. But, in contrast with the equations of Newtonian mechanics, Maxwell's equations turned out not to be invariant under Galilean transformations. Indeed Maxwell insisted that the fundamental equations of electrodynamics were valid only in a unique privileged reference frame, known as the "luminiferous ether." By analogy with waves in gases and elastic solids, it was believed that the electromagnetic waves also needed a medium to propagate. The ether, imagined as an elastic medium permeating all transparent bodies, was assumed to be the carrier of all optical and electromagnetic phenomena. This elastic medium was supposed to provide an absolute reference frame for the electromagnetic phenomena in the same way that Newton's absolute space provided a reference frame for mechanical phenomena. As the ether was only a hypothesis, the need to produce conclusive proof of its existence remained. Because the earth was presumed to move relative to the ether at a certain velocity v, and since the speed of light c relative to the ether was supposed to be constant, it was anticipated that at least sometime during the year the speed of light relative to the earth should be different from c. If a ray of light is reflected from a mirror a distance L away from a light source, and if the source and the mirror are aligned with the direction of motion of the earth relative to the ether, then the time needed for the ray to return to the source has the value
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence, in order to demonstrate the motion of the earth relative to the ether, an apparatus capable of detecting quantities of order (v/c)2 is necessary. In 1881 Michelson performed an experiment by means of an interferometer capable of detecting an effect much smaller than the anticipated one, but he could find no such effect. In 1887 Michelson and Morley repeated the experiment, using a more accurate apparatus, but the experiment again failed to detect the existence of the ether. However, it did demonstrate with a large degree of accuracy that the speed of light is the same in every direction, independent of the motion of the source. In an attempt to explain the experiment's failure to detect the motion of the earth relative to the ether, Fitzgerald advanced a hypothesis according to which a body contracts in the direction of motion. For a rod whose original length in the direction of motion is /0 the length contracts to l= l0 [1–(υ/c)2]½ during motion. This contraction hypothesis was adopted by Lorentz, who generalized it by introducing a set of transformations rendering electromagnetic and optical phenomena independent of uniform motion of the system. In particular, he introduced a variable time, known as the local time because it differs from system to system. This amounted to a dilatation of the time scale. The difference between the Lorentzian and the Galilean transformations is of the order (υ/c)2. Although Lorentz realized that to account for the constancy of the light velocity a new kinematics, namely, the Lorentzian kinematics, was necessary, he did not question the validity of the classical principle of relativity, nor did he abandon the ether theory. In fact, the entire purpose of his transformations was to save the ether concept by providing an explanation for the failure of the experiments to detect uniform motion through the ether.
At about the same time, Poincaré also developed a set of transformations similar to the Lorentz transformation and achieving the same purpose, namely, rendering electromagnetic and optical phenomena independent of uniform motion of the reference frame. Both Lorentz and Poincaré realized that, as a result of these transformations, Maxwell's equations could be expressed in an infinite number of inertial systems, but Lorentz continued to believe that one of these systems represented the ether at rest. Poincaré, however, went one step further by recognizing that the mathematical equivalence of the inertial reference systems for the electromagnetic phenomena represented a new relativity principle. In fact, he proposed this principle as a general law of nature and suggested that the laws of mechanics be modified to conform to this law. However, he never understood the full physical implication of this relativity principle and regarded the transformation purely as a mathematical device. He did not take the important step of making the relativity principle independent of its derivation from the Maxwell equations.
Lorentz and Poincaré made a giant stride toward providing a new description of the physical world, basing that description on sound facts observed by means of reliable experiments rather than basing it on unproved hypotheses. However, they both failed to appreciate the farreaching implications of their transformations. It remained for Einstein to demonstrate that the principle of relativity and the Lorentz transformation raised questions about the very fundamental concepts, such as the ether and absolute space, which were being assumed. Einstein proposed in 1905 to build new principles based on experimental evidence. He advanced two postulates:
1. The laws of nature (including the laws of mechanics and electrodynamics) are the same in all inertial frames.
2. The velocity of light has the same value for all inertial systems, independent of the velocity of the light source.
The two postulates form the basis of Einstein's special theory of relativity. Although the two postulates appear to be contradictory, Einstein showed that they can coexist if the concept of absolute time is discarded and time is added as a fourth coordinate to the three Euclidean spatial coordinates. Einstein did retain the Lorentz transformation, but it must be pointed out that he derived the corresponding equations from the general point of view of the principle of relativity. Later Minkowski concluded that the new Einsteinian kinematics, in which the space and time are inseparable, leads to a new geometrical structure consisting of a fourdimensional space formed by the ordinary space and time. This spacetime world is referred to as world space or Minkowski space. It turns out that the Lorentz transformation is simply the orthogonal transformation of Minkowski space. The new relativity principle does not do away with Galilean inertial frames. On the other hand, by revising the time concept and relating the inertial reference frames by means of Lorentz transformations, Einstein succeeded in providing a common basis for the treatment of both mechanical and electromagnetic phenomena. Overwhelming experimental evidence corroborates the conclusions of special relativity.
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