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2013 | Buch

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The Mathematical Structure of Classical and Relativistic Physics

A General Classification Diagram

verfasst von: Enzo Tonti

Verlag: Springer New York

Buchreihe : Modeling and Simulation in Science, Engineering and Technology

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The theories describing seemingly unrelated areas of physics have surprising analogies that have aroused the curiosity of scientists and motivated efforts to identify reasons for their existence. Comparative study of physical theories has revealed the presence of a common topological and geometric structure. The Mathematical Structure of Classical and Relativistic Physics is the first book to analyze this structure in depth, thereby exposing the relationship between (a) global physical variables and (b) space and time elements such as points, lines, surfaces, instants, and intervals. Combining this relationship with the inner and outer orientation of space and time allows one to construct a classification diagram for variables, equations, and other theoretical characteristics.

The book is divided into three parts. The first introduces the framework for the above-mentioned classification, methodically developing a geometric and topological formulation applicable to all physical laws and properties; the second applies this formulation to a detailed study of particle dynamics, electromagnetism, deformable solids, fluid dynamics, heat conduction, and gravitation. The third part further analyses the general structure of the classification diagram for variables and equations of physical theories.

Suitable for a diverse audience of physicists, engineers, and mathematicians, The Mathematical Structure of Classical and Relativistic Physics offers a valuable resource for studying the physical world. Written at a level accessible to graduate and advanced undergraduate students in mathematical physics, the book can be used as a research monograph across various areas of physics, engineering and mathematics, and as a supplemental text for a broad range of upper-level scientific coursework.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The purpose of this book is to demonstrate the existence of a mathematical structure that is common to all physical theories of the macrocosm and to explain the origin of this common structure. The starting point of this investigation is the analysis of physical variables under a new profile: we take into consideration all those geometric features that are usually overlooked in physics books.

Enzo Tonti

Analysis of Variables and Equations

Frontmatter

Chapter 2. Terminology Revisited

Abstract

Sometimes the same symbol is used with different meanings. Let us look at the following two formulas:

$$\displaystyle{ M =\int \rho \,\, \mathrm{d}V,\qquad W = -\int p\,\,\mathrm{d}V.}$$

(2.1)

In the first integral, M denotes the mass, ρ(P) the mass density at a point P and the symbol dV indicates an infinitesimal volume; in the second integral, W denotes the work, p(V ) the pressure and dV indicates an infinitesimal variation of the volume. In Sect. 5.2 we will distinguish material descriptions from spatial descriptions: in the first integral, V denotes a fixed control volume, typical of a spatial description, whereas in the second integral, V denotes a variable volume, typical of a system description.

Enzo Tonti

Chapter 3. Space and Time Elements and Their Orientation

Abstract

Physics deals with phenomena that arise in space and evolve over time. Our perception of space is based on the existence of bodies and our perception of time is based on the motion of bodies. To describe a body, a device, an instrument, we need to describe its shape and its size, in addition to the materials that compose it. To describe the shape and size of all its parts, we refer to the four space elements, the points, lines, surfaces and volumes. In a spatial description, all physical variables are necessarily associated with one of these four space elements.

Enzo Tonti

Chapter 4. Cell Complexes

Abstract

The role of coordinate systems is to associate to the points of space three numbers, its coordinates. Since we will consider not only points but also lines, surfaces and volumes, we need a reference structure analogous to coordinate systems: these are the cell complexes. They are composed of vertices, edges, faces and cells, i.e. of the four types of space elements. Since a space element can be endowed with one of the two types of orientation, an inner or an outer one, we obtain eight distinct oriented space elements. To represent space elements endowed with an outer orientation, it appears natural to introduce the dual of a cell complex. These eight space elements can be easily organized in a classification diagram.

Enzo Tonti

Chapter 5. Analysis of Physical Variables

Abstract

Physical variables are the foundation of the mathematical formulation of physics because they arise from the existence of quantitative attributes of physical systems. To analyse the mathematical structure of physical theories, it is convenient to start with a detailed analysis of their physical variables. In this chapter we will present two new classifications: the distinction between global variables and their rates and the distinction between configuration, source and energy variables. It is shown that global variables are associated with space elements, and the corresponding densities inherit this association. This analysis was pursued on approximately 180 variables collected in the appendix c.

Enzo Tonti

Chapter 6. Analysis of Physical Equations

Abstract

While physical variables describe the quantitative attributes of physical systems, the equations linking them describe the quantitative behaviour of phenomena, i.e. the physical laws. We will distinguish four kinds of equations used in physics, and for each kind we put into evidence their mathematical structure. The main classification is that of defining equations, topological equations, equations of behaviour and phenomenological equations.

Enzo Tonti

Chapter 7. Algebraic Topology

Abstract

The use of global variables, when combined with a cell complex and its dual, enables the use of algebraic topology. In particular, the notion of cochain, also called discrete form, enables a purely algebraic description of physical fields, and the coboundary process enables a remarkable geometric description of topological laws. The discrete form is the discrete version of the exterior differential forms, and the coboundary operator is the discrete version of the exterior differential on differential forms. In particular the discrete forms on the dual complex, endowed with outer orientation, correspond to the so called twisted differential forms.

Enzo Tonti

Chapter 8. Birth of Classification Diagrams

Abstract

The remarkable association of global physical variables with space and time elements endowed with inner and outer orientations makes it possible to use the same classification diagram of space and time elements to classify global variables and, consequently, their densities and rates, i.e. the field functions. A diagram thus obtained demonstrates the common mathematical structure that underlies classical and relativistic physical theories of the macrocosm.

Enzo Tonti

Analysis of Physical Theories

Frontmatter

Chapter 9. Particle Dynamics

Abstract

In this chapter we provide an operational definition of the variables and equations of particle dynamics. In this presentation, we stress some interesting features which are usually left in the dark. Of particular relevance is the definition of momentum which differs from the one usually given in the literature. Moreover, we analyse in detail the notions of virtual work, potential energy, kinetic energy and kinetic co-energy.

Enzo Tonti

Chapter 10. Electromagnetism

Abstract

In this chapter we give the operational definition of the variables of the electromagnetic field by emphasizing the natural association of global electromagnetic variables with oriented space and time elements. We also present the topological equations, starting directly from the experimental laws, i.e. without passing through the differential formulation. Lastly, we show how to obtain the traditional equations in the differential formulation.

Enzo Tonti

Chapter 11. Mechanics of Deformable Solids

Abstract

In this chapter, we provide an operational definition of variables in the mechanics of deformable solids. In particular, we analyse the global variables to determine their association with space and time elements. We also present the momentum balance without concerning ourselves with the differential formulation. Lastly, we show how to obtain the traditional equations in a differential formulation.

Enzo Tonti

Chapter 12. Mechanics of Fluids

Abstract

In this chapter, we analyse the global variables of fluid dynamics to determine their association with space and time elements. We also present the two major balance equations, the mass and momentum balances, without concerning ourselves with the differential formulation. Lastly, we show how to obtain the traditional equations in a differential formulation.

Enzo Tonti

Chapter 13. Other Physical Theories

Abstract

In thermodynamics a so-called fundamental problem does not exist because the distinction between source and configuration variables need not be introduced. For these reasons the diagrams of thermodynamics differ from the usual diagrams of this book.

Enzo Tonti

Advanced Analysis

Frontmatter

Chapter 14. General Structure of the Diagrams

Abstract

We start with diagram [GEN1], which shows the relation of exterior differential forms with the classification diagram. The physical variables associated with space elements endowed with an inner orientation are described by multicovectors and, hence, by even differential forms. In contrast, the physical variables associated with space elements endowed with an outer orientation are described by pseudo multicovectors and, hence, by odd differential forms.

Enzo Tonti

Chapter 15. The Mathematical Structure

Abstract

Up to now we have analysed the physical and geometrical properties that lead to the discovery of a general structure common to physical theories. These properties lead to building a classification diagram for the variables and the equations of physical theories. We explore now the mathematical properties that arise from this analysis, in particular the algebraic and differential properties of the equations. These properties suggest the need to construct a mathematical model for physical theories.

Enzo Tonti

Backmatter

Titel: The Mathematical Structure of Classical and Relativistic Physics
verfasst von: Enzo Tonti
Verlag: Springer New York
Electronic ISBN: 978-1-4614-7422-7
Print ISBN: 978-1-4614-7421-0
DOI: https://doi.org/10.1007/978-1-4614-7422-7