The Mathematical Structure of Classical and Relativistic Physics (Modeling and Simulation in Science, Engineering and Technology) Sep 07, by Enzo. Enzo Tonti is on Facebook. Join Facebook to connect with Enzo Tonti and others you may know. Facebook gives people the power to share and makes the. The Mathematical Structure of Classical and Relativistic Physics: A General Classification Diagram. Enzo Tonti. from: $ Top of Page. My Account · Billing.

Author: | Doum Migor |

Country: | Ethiopia |

Language: | English (Spanish) |

Genre: | Life |

Published (Last): | 16 January 2016 |

Pages: | 418 |

PDF File Size: | 15.30 Mb |

ePub File Size: | 6.3 Mb |

ISBN: | 693-8-91731-886-8 |

Downloads: | 67978 |

Price: | Free* [*Free Regsitration Required] |

Uploader: | Arashimi |

The Tonti diagram[1] created by the Italian physicist and mathematician Enzo Tonti, is a diagram that classifies variables and equations of physical theories of classical and relativistic physics.

The theories involved are: The classification stems from the observation that each physical variable has a well-defined association with a space and a time element, as shown in Fig.

The starting point of this classification stems from a remark by Maxwell. Speaking about analogies in different physical theories, Maxwell wrote in Of the factors which compose it [energy], one is referred to unit of length, and the other to tonto of area.

This gives what I regard as a very important distinction among vector quantities. This remark leads us to a distinction between vectors referred to lines and vectors referred to surfaces. This is not customarily considered in vector calculus or in physics. In electromagnetism, hence, vectors E and H are associated with lines, and vectors D and B are associated with surfaces.

This distinction leads to a classification of physical variables based on their association with space elements. It is only through the progress of science in recent times that we tonto become acquainted with so large a number of physical quantities that a classification of them is desirable. The classification diagram arises from an analysis of tontj association of physical variables in different physical theories, whatever their mathematical nature, i. Physics, compared to other natural sciences, has the great privilege of considering the quantitative attributes of physical systems and from these originate the physical quantitieswhich can be scalarsvectorstensorsetc.

Physical quantities can be divided into three main categories: The term space-global variables refers to all physical variables that are not densities of other variables.

Distinguishing between global variables and corresponding densities is essential. Mass density is given by the ratio of mass to volume, mass being the global variable. Pressure tonnti the ratio of the normal force on an element of plane surface to the surface area, therefore the normal force is the global variable and the pressure is its surface density. The strain of a bar is the ratio of the elongation to its original length, elongation being the global variable and strain the line density.

The very fact that we create volume, surface or line densities highlights that the corresponding global variables are associated with volumes, surfaces and lines. Every density is obtained by dividing a global variable by the extension of the associated space element.

The mathematical treatment of physics also customarily considers the limit of this ratio to obtain a pointwise density, i. Generally, field variables are preferred to global variables, to the extent that we express the global variables as integrals of the respective field variables, and call them integral variables. Integral variables are global variables, but there are also global variables which are associated with points, and which, therefore, cannot be considered integral variables.

The tonnti time-global variables is used to indicate variables that are not rates of other variables. Displacement, impulse, momentum and position vector are therefore global variables. Displacement and impulse are associated with time intervals, while momentum and position vector are associated with instants.

This is evident if we think that we can perform the rates of the first two variables obtaining velocity and force respectively, while we cannot perform the rates of the last two variables. The physical enzk, both global variables and densities, in turn, can be divided into three classes: Each field has its sources for example, electric charges are the source of electric fields, heat generators are the source of thermal nezo and forces are the source of motion and deformation: The third category, that of energy variablesis created by multiplying a configuration variable by a source variable.

The fundamental problem of a field is finding the configuration of the field at every time instant, given the spatial distribution and the intensity of its sources which is to say: By the term “space elements” we intend pointslinessurfaces and volumeswhich are denoted PLS and V respectively.

By the term “time elements” we intend time instants and time intervalswhich are denoted I and T respectively. Space elements are naturally classified according to their dimension, from the smallest to the biggest, i.

The same classification applies to time elements, i. Lastly all potentials e. A similar association exists between time-global variables and time elements. Therefore, a displacement is associated with a emzo interval Twhereas a position vector is associated with a time instant I.

In this association the measuring process must be taken into account, because temperature is referred to an instant but its measurement involves a time interval – the ennzo needed to reach thermal equilibrium.

Therefore, the temperature is said to be associated with a time interval T. Similarly, the measurement of a force requires a mechanical equilibrium to be reached, hence it is associated with a time interval T. Since in each field of physics the global variables are associated with a space and a time element, the association of the corresponding densities and the rates with the space and time element of the corresponding global variable is spontaneous.

The densities and the rates are said to inherit the association from their corresponding global variables. The same inherited association remains once we have performed the limit to obtain the field variables.

In this classification, the orientation of the space elements has a fundamental role. There are two kinds of orientation: There is a surprising correspondence between the role that a physical variable plays in a theory and the type of orientation of the corresponding space element: To highlight the four space elements, it is expedient to divide a region of space into many cells, thus creating a eno complex.

The vertices represent points, the edges lines, the faces surfaces and the cells themselves volumes. Each of these elements can be given an inner orientation.

Since we also need space elements with outer orientation, it is expedient to consider the dual complexobtained by joining the centroids of the cells belonging to the primal complex. Doing so, we see that the inner orientation of each element of the primal complex induces an outer orientation enzzo the corresponding element of the dual complex. Figures 3 and 4 show how the space elements of the primal complex, which are endowed with inner orientation, tohti an outer orientation on the elements of the dual complex.

Moreover, the space elements of the two complexes can be arranged in a diagram such as the one shown on the right-hand side of the figures.

A cell complex and its dual constitute a valid framework to describe the association of physical variables with the oriented space elements.

To be precise, the configuration variables, which are associated with space elements with inner orientation, may be associated with the elements of the primal complex, while the source variables, which are associated with the space elements with an outer orientation, can be associated with the elements of the dual complex.

At first glance, the notions of inner and outer orientation of a time element may appear strange, certainly unusual. Let us consider a subdivision of the time axis into many adjacent time intervals. Doing so, we have constructed a one-dimensional cell complex in time. Considering now the middle instants of each time interval we have a second cell complex in time.

The first complex is called primal and the second dual. Primal complex, inner orientation: Dual complex, outer orientation: This is called tontk orientation of the dual time gonti. In turn, the eenzo intervals connecting two adjacent dual time instants are called dual time intervals. Once more we can assign them the orientation induced by the primal time instants, as shown in Fig. This is called outer orientation of the dual time intervals.

The fact that physical variables are associated with space and time elements endowed with inner or outer orientation and that space and time elements of a primal and a dual cell complex can be classified as tontu on the right-hand side of Figs.

Let us consider, as an example, gonti classification diagram of electrostatics, shown in Fig. The left column includes those physical variables associated with the primal cell complex, whose elements are endowed with inner orientation: The enoz column includes those physical variables associated with the dual cell complex, whose elements are endowed with outer orientation: The equations linking the variables are contained in the rectangular boxes. The equations in each vertical column are balance equations, circuital equations and gradient-forming equations.

They have a topological structure and are called topological equations. Those that link configuration variables to source variables—contained in the horizontal links—are material or constitutive equations, and belong to the category of phenomenological equations. Particle dynamics, analytical mechanics and electric networks make use of space-global variables that depend only on the time variable. Referring here to particle dynamics, we see that configuration variables are geometrical and kinematic variables, while source variables are static and dynamic variables.

The diagram shows three phenomenological equations: The dissipative nature is shown by the fact that one variable, velocity, changes sign under reversal of motion, while the other, force, does not. Since every global physical variable is associated with both a time enoz a space element, it appears useful to have a classification diagram combining space and time elements.

When the four space elements PLS and V are coupled with the two time elements I and T we obtain eight possible combinations. Furthermore, if one considers that each space element and each time element can be endowed with two orientations, inner and outer, we must combine the 4×2 tnoti space elements with the 2×2 oriented time elements, thus obtaining 32 possible space-time combinations.

The first version associates the space elements endowed with inner orientation with the time elements endowed with inner orientation and vice versa: The second version associates the space elements endowed with inner orientation with the time elements endowed with outer orientation and vice versa. It is the case of field theories such as electromagnetism, gravitation, thermal conduction and irreversible thermodynamics.

The space-time diagrams show clearly which phenomenological equations describe irreversible phenomena: Chronological Tables of famous people. From Wikipedia, the free encyclopedia. This article has multiple issues. Please help improve it or discuss these issues on the talk page. Learn how enzl when to remove these template messages.

This article may fonti too technical for most readers to understand. Please help improve it to make it understandable to non-expertswithout removing the technical details.

February Learn how and when to remove this template message. A major contributor to this article appears to have a close tontj with its subject. It may require cleanup to comply with Wikipedia’s content policies, particularly neutral point of view. Please discuss further on the talk page.