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The Variational Principles of Dynamics cover

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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Contents:
  • Introduction: A Dissection of Compressible Fluid Dynamics
  • The Basic Mathematical Tools:
    • Calculus of Variations
    • Hamiltonian Formalism
    • Hamiltonian Maps
    • Lie Algebras, Generalised Two-Cocycles, Affine Hamiltonian Operators
    • SemiDirect Sum Lie Algebras, Generalised Symplectic Two-Cocycles, Hamiltonian Maps Between Semidirect Sums
  • Abelian Systems (Systems Without Nonabelian Internal Degrees of Freedom):
    • The Prototypical Dynamical Systems and Their Hamiltonian Properties
    • Clebsch Representations (Abelian Case)
    • Variational Principles (Abelian Case)
    • Free Rigid Body
    • Relativistic Compressible Fluid Dynamics
    • Linearization
    • Supervariational Principles (Abelian Case)
  • Nonabelian Systems:
    • Variational Principles of the First Kind
    • Typical Physical Systems
    • Variational Principles of the Second Kind
    • Exceptional Systems

Readership: Mathematical physicists, mathematicians and theoretical physicists.

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FRONT MATTER
  • Pages:i–xviii

https://doi.org/10.1142/9789812798305_fmatter

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INTRODUCTION: A Dissection of Compressible Fluid Dynamics
  • Pages:1–15

https://doi.org/10.1142/9789812798305_0001

Part A - THE BASIC MATHEMATICAL TOOLS


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CALCULUS OF VARIATIONS
  • Pages:19–38

https://doi.org/10.1142/9789812798305_0002

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HAMILTONIAN FORMALISM
  • Pages:39–51

https://doi.org/10.1142/9789812798305_0003

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HAMILTONIAN MAPS
  • Pages:52–57

https://doi.org/10.1142/9789812798305_0004

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LIE ALGEBRAS, GENERALIZED TWO-COCYCLES, AFFINE HAMILTONIAN OPERATORS
  • Pages:58–70

https://doi.org/10.1142/9789812798305_0005

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SEMIDIRECT SUM LIE ALGEBRAS, GENERALIZED SYMPLECTIC TWO-COCYCLES, HAMILTONIAN MAPS BETWEEN SEMIDIRECT SUMS
  • Pages:71–98

https://doi.org/10.1142/9789812798305_0006

Part B - ABELIAN SYSTEMS (SYSTEMS WITHOUT NONABELIAN INTERNAL DEGREES OF FREEDOM)


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THE PROTOTYPICAL DYNAMICAL SYSTEMS AND THEIR HAMILTONIAN PROPERTIES
  • Pages:101–129

https://doi.org/10.1142/9789812798305_0007

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CLEBSCH REPRESENTATIONS (ABELIAN CASE)
  • Pages:130–144

https://doi.org/10.1142/9789812798305_0008

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VARIATIONAL PRINCIPLES (ABELIAN CASE)
  • Pages:145–164

https://doi.org/10.1142/9789812798305_0009

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FREE RIGID BODY
  • Pages:165–172

https://doi.org/10.1142/9789812798305_0010

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RELATIVISTIC COMPRESSIBLE FLUID DYNAMICS
  • Pages:173–180

https://doi.org/10.1142/9789812798305_0011

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LINEARIZATION
  • Pages:181–221

https://doi.org/10.1142/9789812798305_0012

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SUPERVARIATIONAL PRINCIPLES (ABELIAN CASE)
  • Pages:222–303

https://doi.org/10.1142/9789812798305_0013

Part C - NONABELIAN SYSTEMS


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VARIATIONAL PRINCIPLES OF THE FIRST KIND
  • Pages:307–325

https://doi.org/10.1142/9789812798305_0014

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TYPICAL PHYSICAL SYSTEMS
  • Pages:326–379

https://doi.org/10.1142/9789812798305_0015

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VARIATIONAL PRINCIPLES OF THE SECOND KIND
  • Pages:380–392

https://doi.org/10.1142/9789812798305_0016

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EXCEPTIONAL SYSTEMS
  • Pages:393–401

https://doi.org/10.1142/9789812798305_0017

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BACK MATTER
  • Pages:403–422

https://doi.org/10.1142/9789812798305_bmatter

“This book yields a self-contained, rigorous but also very clearly written account of the subject … the book also contains a 'wealth' of apparently carefully selected and often very amusing citations from various sources.”
Mathematical Reviews