Classical Dynamics

A Contemporary Approach

by Jorge V. José and Eugene Saletan
Classical Dynamics Bood

Chapter Nine:

Continuum Dynamics






Fig. 8.16 small
Lagrangian formulation of continuum dynamics
  1. 1. passing the continuum limit: the sine-Gordon equation, the wave and Klein-Gordon equations
  2. 2. the variational principle: introduction, variational derivation of the EL equations, the functional derivative, discussion
  3. 3. Maxwell's equations: some special relativity, electromagnetic fields, the Lagrangian and the EL equations

Noether's theorem and relativistic fields
  1. 1. Noether's theorem: the theorem, conserved currents, energy and momentum in the field, example: the electromagnetic energy-momentum tensor
  2. 2. relativistic fields: Lorentz transformations, Lorentz invariant L and conservation, free Klein-Gordon fields, complex K-G field and interactionwith the Maxwell field, discussion of the coupled field equations
  3. 3. spinors: spinor fields, a spinor field equation

Hamiltonian formalism
  1. 1. the Hamiltonian formalism: definitions, the canonical equations, Poisson brackets
  2. 2. expansion in orthonormal functions: orthonormal functions, particle-like equations, example: Klein-Gordon

Nonlinear field theory
  1. 1. sine-Gordon equation: soliton solutions, properties of sG solitons, multiple-soliton solutions, generating soliton solutions, nonsoliton solutions, Josephson junctions
  2. 2. nonlinear KG equation: the Lagrangian and the EL equation, kinks

Fluid Dynamics
  1. 1. The Euler and Navier-Stokes equations: substantial derivative and mass conservation, Euler's equation, viscosity and incompressibility, the Navier-Stokes equations, turbulence
  2. 2. The Burgers equation: the equation, asymptotic solution
  3. 3. Surface waves: equations for the waves, linear gravity waves, nonlinear shallow water waves: the KdV equation, single KdV solitons, multiple KdV solitons

Hamiltonian formalism for nonlinear field theory
  1. 1. the field theory analog of particle dynamics: from particles to fields, dynamical variables and equations of motion
  2. 2. the Hamiltonian formalism: the gradient, the symplectic form, the condition for canonicity, Poisson brackets
  3. 3. the KdV equation: KdV as a Hamiltonian field, constants of the motion, generating the constants of the motion, more on constants of the motion
  4. 4. the sine-Gordon equation: two-component field variables, sG as a Hamiltonian field

Problems
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