Variational Principles of Continuum Mechanics

1. Front Matter

   Pages i-xiv

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2. Fundamentals

   1. Front Matter

      Pages 1-1

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   2. Variational Principles

      • V.L. Berdichevsky

      Pages 3-44

   3. Thermodynamics

      • V.L. Berdichevsky

      Pages 45-65

   4. Continuum Mechanics

      • V.L. Berdichevsky

      Pages 67-115

   5. Principle of Least Action in Continuum Mechanics

      • V.L. Berdichevsky

      Pages 117-147

   6. Direct Methods of Calculus of Variations

      • V.L. Berdichevsky

      Pages 149-282

3. Variational Features of Classical Continuum Models

   1. Front Matter

      Pages 283-283

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4. Variational features of classical continuum models

   1. Statics of a Geometrically Linear Elastic Body

      • V.L. Berdichevsky

      Pages 285-339

   2. Statics of a Geometrically Nonlinear Elastic Body

      • V.L. Berdichevsky

      Pages 341-374

   3. Dynamics of Elastic Bodies

      • V.L. Berdichevsky

      Pages 375-387

   4. Ideal Incompressible Fluid

      • V.L. Berdichevsky

      Pages 389-454

   5. Ideal Compressible Fluid

      • V.L. Berdichevsky

      Pages 455-472

   6. Steady Motion of Ideal Fluid and Elastic Body

      • V.L. Berdichevsky

      Pages 473-494

   7. Principle of Least Dissipation

      • V.L. Berdichevsky

      Pages 495-508

   8. Motion of Rigid Bodies in Fluids

      • V.L. Berdichevsky

      Pages 509-529

5. Back Matter

   Pages 1-53

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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.

• Calculus of Variations
• Degrees of freedom
• Dissipation
• Entropy
• Variational Principles
• calculus
• continuum mechanics
• mechanics
• statics
• thermodynamics
• fluid- and aerodynamics

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)

• Mechanical Engineering Dept., Wayne State University, Detroit, U.S.A.

  Victor Berdichevsky

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