Variational Principles of Continuum Mechanics: I. Fundamentals (Interaction of Mechanics and Mathematics) 2010th Edition by Victor Berdichevsky (PDF)

Description

Thereareabout500booksonvariationalprinciples. Theyareconcernedmostlywith the mathematical aspects of the topic. The major goal of this book is to discuss the physical origin of the variational principles and the intrinsic interrelations between them. For example, the Gibbs principles appear not as the rst principles of the theory of thermodynamic equilibrium but as a consequence of the Einstein formula for thermodynamic uctuations. The mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for direct study of variational problems. Thebookisacompletelyrewrittenversionoftheauthor’smonographVariational Principles of Continuum Mechanics which appeared in Russian in 1983. I have been postponing the English translation because I wished to include the variational pr- ciples of irreversible processes in the new edition. Reaching an understanding of this subject took longer than I expected. In its nal form, this book covers all aspects of the story. The part concerned with irreversible processes is tiny, but it determines the accents put on all the results presented. The other new issues included in the book are: entropy of microstructure, variational principles of vortex line dynamics, va- ational principles and integration in functional spaces, some stochastic variational problems, variational principle for probability densities of local elds in composites with random structure, variational theory of turbulence; these topics have not been covered previously in monographic literature.

User’s Reviews

Editorial Reviews: Review From the reviews:“This new book goes far beyond anything currently available concerning variational principles in continuum mechanics. … We have at hand a monument that all students and professionals in applied mathematics physics and engineering will praise and naturally keep handy on their bookshelf. Teachers will find in the book a wealth of pedagogical material for many one semester courses. They and their students will appreciate the clarity simplicity and ingenuity of many arguments offered without pedantry and sacrifice of rigour.” (Gerard A. Maugin, Mathematical Reviews, Issue 2011 a) From the Back Cover The book reviews the two features of the variational approach: its use as a universal tool to describe physical phenomena and as a source for qualitative and quantitative methods of studying particular problems.Berdichevsky’s work differs from other books on the subject in focusing mostly on the physical origin of variational principles as well as establishing their interrelations. For example, the Gibbs principles appear as a consequence of the Einstein formula for thermodynamic fluctuations rather than as the first principles of the theory of thermodynamic equilibrium. Mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for the direct study of variational problems. In addition, a thorough account of variational principles discovered in various branches of continuum mechanics is given.In this book, the first volume, the author covers the variational principles for systems with a finite number of degrees of freedom; the variational principles of thermodynamics; the basics of continuum mechanics; the variational principles for classical models of continuum mechanics, such as elastic and plastic bodies, and ideal and viscous fluids; and direct methods of calculus of variations.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book will be a classic among books on variational principles of mechanics. As compared with another bestseller book written by Lanczos it contains many new developments of the variational principles and their applications in recent years. Among these new developments I would put on the first places the variational principles of irreversible processes and the variational-asymptotic method. The other new topics included in the book compared with the Russian edition are: entropy of microstructure, variational principles of vortex line dynamics, variational principles and integration in functional spaces, some stochastic variational problems, variational theory of turbulence.

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