Variational Principles of Continuum Mechanics: I. Fundamentals (Interaction of Mechanics and Mathematics)

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. 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. Front Matter....Pages i-xiv Front Matter....Pages 1-1 Variational Principles....Pages 3-44 Thermodynamics....Pages 45-65 Continuum Mechanics....Pages 67-115 Principle of Least Action in Continuum Mechanics....Pages 117-147 Direct Methods of Calculus of Variations....Pages 149-282 Front Matter....Pages 283-283 Statics of a Geometrically Linear Elastic Body....Pages 285-339 Statics of a Geometrically Nonlinear Elastic Body....Pages 341-374 Dynamics of Elastic Bodies....Pages 375-387 Ideal Incompressible Fluid....Pages 389-454 Ideal Compressible Fluid....Pages 455-472 Steady Motion of Ideal Fluid and Elastic Body....Pages 473-494 Principle of Least Dissipation....Pages 495-508 Motion of Rigid Bodies in Fluids....Pages 509-529 Back Matter....Pages 1-53 This is a concise and understandable book about variational principles of continuum mechanics. The book is accessible to applied mathematicians, physicists and engineers who have an interest in continuum mechanics