Variational Principles of Continuum Mechanics: I. Fundamentals

Victor Berdichevsky (auth.)9783540884668, 3540884661

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.

Table of contents :
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