Хмельник Соломон Ицкович : другие произведения.

Variational Principle of Extremum in Electromechanical and Electrodynamic Systems

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  • Аннотация:
    Here we shall formulate and prove the variational optimum principle for electromechanical systems of arbitrary configuration, in which electromagnetic, mechanical, thermal, hydraulic or other processes are going on. The principle is generalized for systems described by partial differential equations, including also Maxwell equations. The presented principle permits to expand the Lagrange formalism and extend the new formalism on dissipative systems. It is shown that for such systems there exists a pair of functionals with a global saddle point. A high-speed universal algorithm for such systems calculation with any perturbations is described. This algorithm realizes a simultaneous global saddle point search on two functionals. The algorithms for solving specific mathematical and technical problems are cited. The book contains numerous examples, including those presented as M-functions of the MATLAB system and as functions of the DERIVE system.
  
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  Contents
  
  Detailed Contents \ 5
  Preface \ 11
  Chapter 1. RCL-circuits with Electric Charges \ 16
  Chapter 2. RCL-circuits with Electric Currents \ 32
  Chapter 3. Special Transformers in Alternating Current Circuits\ 42
  Chapter 4. Generalized Functional \ 55
  Chapter 5. Electric Circuit Computing Algorithms \ 62
  Chapter 6. Variational Principle and Maximum Principle \ 102
  Chapter 7. Electromechanical Systems \ 145
  Chapter 8. The Functional for Partial Differential Equations \ 156
  Chapter 9. The Functional for Maxwell Equations \ 191
  Сhapter 10. Principle extremum of full action \ 324
  References \ 346
  Main Notations \ 350
  Some of the Terms \ 353
  
  Preface
  
  The search for variational principles for the electromechanical systems of arbitrary structure and configuration is a subject of theoretical and practical interest. In this connection we shall consider below a problem of looking for such a functional whose steady-state equations are equations of electromechanical system. For mechanical systems such principles are generally known. For special cases of electric circuits the solution of this problem is known. For instance, for circuits with resistances the solution has been found by Maxwell [1], and was extended not long ago to circuits with diodes and direct-current transformers [2]. Another generalization for circuits with non-linear resistances may be found in [3, 4]. For circuits with capacitances and inductances (but without resistances) is also known [3, 5]. In [6] the works are listed in which attempts were made to solve the problem for general-form electric circuit, and all these attempts were proved insolvent. The reason for such search is understandable, as the absence of extremum principle for electric circuits seems to be rather strange. As regards to the practical side of the question, the existence of such principle permits to use alternate current electric circuits for calculus of variation problems simulation: these circuits are nature"s own simple-device computer that solves a very complicated mathematical problem (using an algorithm of unknown kind).
  On the other hand, a discussion in the terms of electric circuits may lead to the development of certain problems of calculus of variations. An example of a similar influence of the direct-current electric circuits theory on the theory of mathematical programming may be found in the work [2]. Lastly, the calculus of variations theory may also be used for electric circuits and electromechanical systems computing. Such approach has been used by the author. The extremum principle for alternating current electric circuit was formulated by the author in 1988 in [8] and was developed in the articles [9, 10, 15, 16]. The first edition of this book was published in [31].
  The basic idea is that the current function is "split" into two independent functions. The proposed functional contains such pairs of functions; its optimum is a saddle point, where one group of functions minimizes the functional, and the other one - maximizes it. The sum of the optimal values of these functions gives the current function of the electric circuit.
  The previously presented results will be generalized and developed below; the computational aspect of this principle"s use will be considered as well. Furthermore, this principle will be extended to electromechanical systems, since it may be integrated with a principle known in mechanics as the minimal action principle, since it is a generalization of a known principle of least action. For a given electromechanical system a functional containing functions of thermal, mechanical, electric and electromagnetic energies, as well as the functions describing the perturbation actions - electric and mechanical, is formed. These functions depend on the system"s configuration. The functional has the dimension: "energy*time". The functional is a quadratic function of the sought parameters, and it has a sole optimal point. There are no constraints (they are also included into the functional). The functions providing the optimal value of the functional present solution of the given electromechanical system"s calculation problem. Consequently, the given electromechanical system"s calculation may be stated mathematically as a variational problem of seeking an unconditional optimum of a quadratic functional. Such problem always has a solution, and a fast algorithm has been found for the search of this functional"s saddle point.
  The described principle may be used for the development of a universal package of programs for fast computation of arbitrarily structured and configured electromechanical systems.
  
  So, the nature gives us by the said principle a certain functional. The second Kirchhoff"s law equations follow from the optimization of this functional with constraints in the form of the first Kirchhoff"s law equations. So naturally the optimization of the said functional and the solution of the system of Kirchhoff"s law equations both lead to the same result.
  The proposed method extends to partial differential equations, including also the Maxwell equations.
  In essence we are presenting a generalization of a known Lagrange formalism - an universal method of physical equations derivation from the least action principle. However, the Lagrange formalism is applicable only to those systems where the full energy (the sum of kinetic and potential energies) is kept constant. It does not reflect the fact that in real systems the full energy (the sum of kinetic and potential energies) decreases during motion, turning into other types of energy, for example, into thermal energy , i. e. there occurs energy dissipation. Thus, the presented formalism is extended on dissipative systems.
  
  The book consists of 9 chapters.
  In Chapter 1 the electric circuit with RCL-elements is considered and a functional from the split function of charges x and y is formulated for this circuit. It is shown that the said functional it maximized as a function of x and minimized as a function of y. The sum of the optimal values of x and y is equal to the observed function of charges q. A computational method of searching for the functional"s saddle point is presented.
  In Chapter 2 the extremum principle for functional of split function of currents v and w is similarly considered. It is shown that the said functional is being maximized as the function of v and minimized as the function of w. The sum of the optimal values of v and w is equal to the observed function of currents g. A computing method of searching for the functional"s saddle point is presented.
  In Chapter 3 the electric circuits are supplemented by instanteous current values transformers. Such transformers were originally explored by Dennis and will in future be called Dennis transformers. It is shown that in this case for electric circuit there also exist functionals from split functions of charge and of current. The first Kirchhoff"s law equations serve as constraints in the search of saddle point for these functionals. The existence of second Kirchhoff"s law equations follow from the existence of saddle points for these functionals. Then the circuits are modified in such a way that they become mathematically equivalent to simple RCL-circuits and may be described by functionals without constraints. The calculation of such circuits (called unconstrained) becomes significantly simpler. Then we shall consider the so called integral transformers and circuits containing them. These transformers present a certain generalization of Dennis transformers, and in sinusiodal current circuits they are equivalent to transformers with a complex turn ratio.
  In Chapter 4 a method is proposed for finding such functions of charges and currents, that their optimal values provide the optimum of the two functionals simultaneously. The physical interpretation of the functionals is considered, and it is shown that in the electric circuit the influence of thermal and electromagnetic energy is optimized simultaneously.
  In Chapter 5 the algorithms of simultaneous optimization of the said functionals are described. The most commonly encountered types of voltage and current sources are considered as the functions of time - sinusoidal, periodical and step functions. The same functions may be viewed as permutation actions in a system of differential equations, whose solution amounts to the electric circuit calculation with the aid of the proposed method. It is shown that the solution of linear algebraic equation system also amounts to calculation of an electric circuit with sinusoidal currents, using the suggested method.
  Chapter 6 discusses some concepts of the Pontryagin"s maximum principle. It is shown that this principle may be used for the electrical circuit functional optimization. Thereby it is established that the considered variational principle may be extended also for discontinuous functions. This argument was used above in the description of discontinuous functions calculation method. Further we shall describe an algorithm of electrical circuit calculation, based on the combination of variational principle and maximum principle.
  In Chapter 7 we consider the analogy between the presented and the Lagrange formalism. Then we turn to the discussion of electromechanical systems. The electric circuit is complemented by some electromechanical elements, which involve, along with currents and charges, some "foreign" variables, such as coordinates, velocities, accelerations, forces, moments, temperature, pressure etc, describing the non-electric processes - mechanical, thermal, hydraulic. A system of equation is built, describing a system of electromechanical elements, connected into an electric circuit. It is shown that such system of equations is also equivalent to the conditions of existence of two functionals, similar to the functionals for electric circuits. The optimum principle for these functionals in some particular cases is transformed into the principle of minimal action.
  In Chapter 8 we are dealing with electric circuits, which are described by partial differential equations - electric lines, planes, volumes. We consider classic and special partial differential equations. We show that for them it is also possible to build functionals, and the search for these functionals extremum is equivalent to the solution of these equations.
  In Chapter 9 it is proved that there exists a functional for which Maxwell equations are the necessary and sufficient conditions of global extremum existence, and this extremum is a saddle point. The subject is the computational aspect which is illustrated by detailed examples of computations for various electromagnetic fields. The method allows to formulate and to solve the sort of Maxwell equations systems that have solutions with unusual physical interpretation:
   longitudinal electromagnetic waves,
   standing waves in the absence of energy exchange between the electric and magnetic component
   electric waves in the absence of magnetic waves and vice versa.
  
  In Chapter 10 we present a new variational extremum principle of general action, which extends the Lagrange formalism to dissipative systems. We show that this principle is applicable to electrical engineering, mechanics with regard to friction, electrodynamics and hydrodynamics. The prove is in the results stated in the previous chapters. The proposed variational principle is a new formalism, which permits to build a functional with one optimum saddle line for various physical systems. Moreover, the new formalism is not only universal method of deducing physical equations from a certain principle, but also a computational method for these equations.
  
  The book includes numerous examples. Part of them are M-functions of the MATLAB system. These programs comprise a significant part of the book, as a part of computational formulas is simply included into the programs. It was possible because MATLAB language is nearly as laconic as traditional mathematical language, particularly in the part concerned with operations with vectors and matrices which are being widely used in this book.
  
  The book is accompanied by the disk CD "Programs for electric circuits calculation in the MATLAB systems". The disk may be purchased separately - see http://www.lulu.com/content/778812. The disk contains open codes of numerous MATLAB programs, including also those functions which are given in the examples directly in the book's text. In these examples, in their turn, there are references to files on the disk, which contain the functions from the examples and tests for their testing. The tests are presented in a form enabling their use for practical computations.
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