The Variational Principles of Dynamics

By: BORIS A KUPERSCHMIDT

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Hardcover | 1 December 1991

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442 Pages

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22.86 x 15.24 x 2.54

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Given a conservative dynamical system of classical physics, how does one find a variational principle for it? Is there a canonical recipe for such a principle? The case of particle mechanics was settled by Lagrange in 1788; this text treats continuous systems. Recipes devised are algebraic in nature, and this book develops all the mathematical tools found necessary after the minute examination of the adiabatic fluid dynamics in the introduction. These tools include: Lagrangian and Hamiltonian formalisms, Legendre transforms, dual spaces of Lie algebras and associated 2-cocycles; and linearized and Z₂-graded versions of all of these. The following typical physical systems, together with their Hamiltonian structures, are discussed: Classical Magnetohydro-dynamics with its Hall deformation; Multifluid Plasma; Superfluid He-4 (both irrotational and rotating) and 3He-A; Quantum fluids; Yang-Mills MHD; Spinning fluids; Spin Glass; Extended YM Plasma; A Lattice Gas. Detailed motivations, easy-to-follow arguments, open problems, and over 300 exercises help the reader.

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Preface
Acknowledgements
Xenagogue
Introduction: A Dissection of Compressible Fluid Dynamics	p. 1
The Basic Mathematical Tools
Calculus of Variations	p. 19
Polynomial Lagrangians	p. 20
Nonpolynomial Lagrangians	p. 36
Hamiltonian Formalism	p. 39
Hamiltonian Maps	p. 52
Lie Algebras, Generalized Two-Cocycles, Affine Hamiltonian Operators	p. 58
Semidirect Sum Lie Algebras, Generalized Symplectic Two-Cocycles, Hamiltonian Maps between Semidirect Sums	p. 71
Semidirect Sum Lie Algebras	p. 72
Symplectic Two-Cocycles	p. 85
Hamiltonian Maps between Semidirect Sums	p. 93
Abelian Systems (Systems Without Nonabelian Internal Degrees of Freedom)
The Prototypical Dynamical Systems and Their Hamiltonian Properties	p. 101
Magnetohydrodynamics	p. 102
Multifluid Plasma	p. 111
Superfluid Helium-4	p. 119
Clebsch Representations (Abelian Case)	p. 130
Clebsch Representations: Cocycleless Case	p. 130
Clebsch Representations in the Presence of Symplectic Two-Cocycles	p. 138
Variational Principles (Abelian Case)	p. 145
Variational Legendre Transformation	p. 146
The Construction of Constrained Lagrangians (Cocycleless Case)	p. 150
Constrained Lagrangians in the Presence of Symplectic Two-Cocycles	p. 155
Free Rigid Body	p. 165
Relativistic Compressible Fluid Dynamics	p. 173
Linearization	p. 181
Calculus	p. 182
General Hamiltonian Objects	p. 190
Lie-Algebraic Objects	p. 199
Abelian Clebsch Representations	p. 205
Legendre Transforms	p. 212
Variational Principles	p. 215
Supervariational Principles (Abelian Case)	p. 222
Z[subscript 2]-Graded Calculus	p. 222
SuperHamiltonian Formalism	p. 241
Lie Superalgebras	p. 253
SuperClebsch Representations	p. 270
Z[subscript 2]-Graded Legendre Transformations	p. 277
Z[subscript 2]-Graded Constrained Lagrangians: Even Case	p. 285
Z[subscript 2]-Graded Constrained Lagrangians: Odd Case	p. 294
Nonabelian Systems
Variational Principles of the First Kind	p. 307
Clebsch Representations of the First Kind	p. 308
Constrained Lagrangians of the First Kind	p. 315
Super Versions	p. 321
Typical Physical Systems	p. 326
Irrotational Superfluid Helium-4	p. 326
Rotating Superfluid Helium-4	p. 332
Anisotropic Spinless Superfluid [superscript 3]He-A	p. 343
Quantum Fluids	p. 346
Yang-Mills Plasma	p. 348
Yang-Mills Magnetohydrodynamics	p. 351
Momentum Conservation	p. 354
Spinning Fluid	p. 357
Hall Magnetohydrodynamics	p. 362
Variational Principles of the Second Kind	p. 380
Unbinding Map	p. 380
[actual symbol not reproducible] {Clebsch Representations}	p. 384
Sniper Versions	p. 389
Exceptional Systems	p. 393
Spin Class	p. 393
Extended Yang-Mills Plasma	p. 397
A Lattice Gas	p. 399
Sources and Comments	p. 405
Symbols	p. 411
Bibliography	p. 417
Table of Contents provided by Blackwell. All Rights Reserved.

The Variational Principles of Dynamics

At a Glance

Hardcover

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