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The exhaustive list of topics in Classical Field Theory in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Hamiltonians and Lagrangians. Legendre transforms and their properties. Euler-Lagrange equations. Principle of least action.
  • Group theory from invariances of classical equations.Newton's equations and the Galilean group. Maxwell's equations. Special Relativity and the Lorentz group. Vectors and tensors of the rotation and Lorentz groups.
  • Systems with infinite degrees of freedom. Locality in space and time. Lagrangian densities for real and complex scalar fields. Euler-Lagrange (EL) equations. Functional calculus revisited. Hamiltonian density. The energy-momentum tensor.
  • Finite-energy time-independent solutions -- classical vacua. Kinks in the Sine-Gordon and f4 theories. Green functions as singular solutions. Boundary conditions.
  • Discrete and continuous symmetries. Noether's theorem: the energy momentum tensor, the generalized angular momentum and the electromagnetic current.
  • Lagrangian density. Gauge invariance and the electromagnetic field strength. Maxwell's equations. Lorentz invariants of the field strength. The symmetrized energy-momentum tensor. The generalized angular momentum and the spin of the photon.
  • Global symmetries. Spontaneous breakdown of symmetry. Goldstone's theorem.
  • Solitons as finite-energy solutions. Derrick's theorem. Getting around Derrick's theorem. Local symmetries and gauge fields. Abelian vortices. The Dirac monopole as a singular solution of Maxwell's equations. Dirac quantization.
  • Abelian gauge fields. Covariant derivatives and minimal coupling. The abelian Higgs model. Vortex solutions (in type II superconductors). Topological conservation laws. The abelian Higgs mechanism.
  • Covariant derivatives; The Yang--Mills field strength; Coupling matter to non-abelian gauge fields; Higgs mechanism - SU(2)->U(1) and SO(3)->U(1) ; Weinberg's theorem. The 't Hooft-Polyakov monopole as a non-singular solution. Julia-Zee dyons; the Bogomolnyi-Prasad- Sommerfield(BPS) limit. Dirac quantization for dyons.
  • Symmetry breaking in the electroweak sector. Quantum Chromodynamics and the quark model.
  • Instantons as finite action solutions to the EL equations.The 't Hooft solution. Nahm and Bogomolnyi equations from dimensional reduction.