by Boris A Kupershmidt (The University of Tennessee Space Institute)

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.

• 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