Classical Mechanics
An Overview
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Note that in the context of gravitation, what is observable is the force that exists when there are two masses present. The following hypothetical experiment allows for a detailed description of the process by which we proceed from observable forces to non-observable fields. Consider two masses, m and m’: m will be left unmolested during the process; m’ will be used as a test mass and will be allowed to vary. When m’ is introduced into the region proximate to m, it is subject to an attractive force which is measured as F = -G mm'/r^2 where r is the distance between the (centers of the) masses. Now, if m’ is moved to another location with respect to m, a force different in both magnitude and direction is observed and found still to be in accordance with the above equation. Then at another location, then another, etc. It can be construed from the fact that the force vectors can be determined from the above equation and that no force is observed when the mass m is removed, that m is the cause of these forces and that its existence somehow predisposes the space in its vicinity. If we are to pursue this idea, we are faced with the task of quantifying the preconditioning of the space by m and it must be quantified in such a way that it is independent of m’. We note first that F depends on the size of m’. We attempt to eliminate this aspect of the problem by considering F', the force per unit m’: F^' = F/m' . Next we argue that the very presence of m’ may be altering the condition of the local space, so we try to minimize any possible effect by letting m’ become smaller and smaller, presuming that if it is sufficiently small, it can’t be screwing things up too much. Mathematically, this is taking the limit as m’→0. We conclude at this point that what we have is something which depends only on m and since this ‘something’ extends throughout the region of space near m we call it a field, and since it is associated with the force of gravity, we call it the gravitational field produced by the mass m. Let U represent this gravitational field produced by m. Then the above description in mathematical symbols is U = 〖lim〗┬(m^'→0)〖F^' 〗 = 〖lim〗┬(m^'→0)〖 F/m'〗 U = -G m/r^2 〖 〖lim〗┬(m^'→0)〗〖 m'/m'〗 U = -G m/r^2 . This was the argument intended to justify the existence of the electrostatic field and although over a century late, it was also used to justify the existence of the gravitational field. The field U quantifies the presumed preconditioning of the space in the vicinity of m. It is contrived in such a way that if an arbitrary second mass m’ is introduced into the region, then the force acting on it is F = m'U Do you buy it?? ….. Well don’t! 6.3.2 The Emperor’s New Clothes With pomp and ceremony did the emperor parade naked through the streets of the town, and everyone believed himself unworthy to see his magical clothes which, they were told, would be visible only to those who were without blemish. Then a child in all innocence turned to his mother and said, “The emperor has no clothes!” and then … Classical field theory parades similarly. Its tailors have dressed it in a logic that isn’t there, embellished it with mathematical finery which does not fit and cloaked it in a robe of mystery behind which it simply cannot hide. Let’s examine the definition of the previous section a little more closely. The issue of logic enters at the beginning of the section with the statement, “…can be construed … that m is the cause of these forces…”. The fact that an existing condition can be construed as implying the existence of a field is not compelling enough logically to instate such a field as a physical reality. A metaphysical conclusion about existence or non-existence requires more than construing. A lengthy discussion of this point was presented in section 3.3.1. A second issue concerning the logic is the limiting process which is used to define the classical field. Its conclusion follows from a premise that pretends to be logical but isn’t: namely, it assumes that ‘almost not there’ is almost the same as ‘not there’. Indeed, the concept of ‘almost not there’ in itself is a little strange, if not downright comical. But, delivered with the proper pomp and ceremony, one could easily believe himself unworthy to comprehend the mystery at hand. It should also be noted that the gravitational force is related to m’ as a simple proportion. When the force was divided by m’, to find the force per unit m’, the result was already independent of m’. The subsequent limiting process is, in the case in point, totally ineffectual. It stands as a pseudo-philosophical argument to convince us that the field is really there. In the final analysis, we have simply divided out the troublesome m’; any subsequent embellishment such as the limiting process attempts to justify and/or mask this embarrassing fact. . . So indeed, the emperor has no clothes…but… … he is still the emperor. The tailor and those in attendance who raved about his clothes may have unresolved business with the emperor and the emperor himself may have some gullibility issues, but he still reigns. And so it is with fields. It is not the point here to discredit fields and the immensity of their power in the description of physical situations; the purpose is to instill a clear understanding of their relationship to physics. The definition of classical fields establishes them as an extremely useful mathematical device but not as physical realities. It is of great importance to make use of these alleged fields in the analyses of physical systems; it is equally important to know that they have not been established as realities. (See section 3.3.1.1.)
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About the Book
For the most part, the book presents the same material that is usually covered in a typical first course in mechanics. There are, however, several noteworthy exceptions to this where the material presented here reaches beyond this boundary. Among these are the material on the dot and cross products of vectors, the analysis of the two-body problem, the discussion of flux and Gauss’s theorem, the calculation of particular gravitational field configurations, and the philosophical assertions about the existence of fields. These topics are basic to the nature of physics and its applications and, as such, must be addressed early on in any cohesive endeavor to understand the context of physical thinking whether it be classical or modern. They are included here so that they may assume their proper place at the foundation of what might be called the structure of one’s thoughts about physics. The material is presented in an order that accommodates an axiomatic approach using Newton’s laws as the axioms. The book then proceeds to those analyses that follow most simply from them. This process leads easily and naturally to the definitions of such quantities as momentum, energy, impulse, work, etc. These are the quantities natural to Newtonian mechanics and consequently become the parameters most universally used to describe systems that lend themselves to a Newtonian analysis. As the book develops, it makes use of these basic concepts to address more complex issues such as circular motion, torque, combined translational and circular motion, etc. The book is written in an informal lecture style and is focused on the understanding of Newtonian mechanics rather than on developing a prowess in problem solving. Its uniqueness is difficult to describe as it is laced intractably throughout its pages and derives from the author’s ability to cast each topic in the context of the simple manifestation of an understandable underlying principle.
About the Author
Mr. Sparapany attended MIT on a partial scholarship, where he received a bachelor’s degree in physics in 1957. During his twelve years as a member of this community, much of his time was spent teaching physics and mathematics to undergraduates. The consensus of those who are associated with him in this context is that he has a knack for clarifying and simplifying issues often considered to be complex and elusive, imparting to his students an understanding of the subject material that is both deep and foundationally solid. This type of feedback pervades his entire teaching career and, in fact, was a major factor in prompting the writing of this book. After leaving the MIT community, Mr. Sparapany’s endeavors were focused on a number of diverse scientific issues extant at the time. These included the design and construction of special microwave devices for use in communications and radar detection and the understanding of spoken language by computer. There were two projects he addressed during that time that were of particular interest because they required an original cutting-edge approach to their mathematical description: 1. The removal of the contamination introduced into telescopic seeing by the strain in the reflecting mirror caused by its own weight 2. The analysis of entropy-based decision criteria and their application to adaptive pattern recognition and artificial intelligence The first of these is one particular next step in the older problem of improving telescopic imagery; the second represents a theory by which a machine (computer) starting with certain abilities can improve them as it operates or, stated a little differently, the theoretical analysis of a process by which a computer learns from its experience. But Mr. Sparapany is a jack-of-all-trades—he also dabbles in painting (see the back cover of this book), he is an accomplished pianist who has performed and taught piano, and he composes sacred choral music and has worked both as an accompanist and a church organist. Additionally, he has written two math books. One is a brief introduction to linear analysis, which deals with the algebra of matrices and vectors. The other is a complete solution to Pell’s equation together with an exposition of some topics related to it. On the more offbeat side, he has written the story for a movie, which satirizes the antics involved in the machinations of the Cold War. And finally, he has also written a children’s story.