Topics

• Magnetic and electric fields

• The concept of field

• The “del” or “nabla” symbol

• Vector calculus: Gradient, Divergence and Curl

• The Levi-Civita symbol

• Algebra: div curl and curl grad vanish

• The vector potential and why it’s needed

• Gauge field: “Gauge” is a misnomer

• Lorentz force. Lorentz force compared with the Coriolis force

• Lagrangian for charged particles in a electro-static and magneto-static fields

• Gauge invariance of the equations of motions associated to the electro-magneto-static Lagrangian

References

• Field theory

• Lorentz force.

• Electromagnetism

• Vector calculus

• Levi-Civita symbol

Topics

• Review of the vector potential, concept of gauge and gauge invariance

• Lorentz force law

• Example of different vector potentials for a constant magnetic field and the gauge transformation that relate them

• Importance of gauge invariance and choice of gauge

• Lagrangian of a particle in a static magnetic field. Review of the related action gauge invariance

• Distinction between mechanical and canonical momentum: only the canonical momentum is related to symmetries and invariance

• Derivation of the Euler-Lagrange equation of motion from the magneto-static Lagrangian and rediscovery of the Lorentz force

• Justification of the vector potential as an essential tool for the least action principle

• Derivation of the magneto-static Hamiltonian

• Smart choice of gauge and derivation of the Lorentz force from symmetry arguments only, “cyclic coordinates”

• Circular motion of a charged particle in a static magnetic field

• Monopoles discussion as part of the questions session

• Brief Quaternions discussion as part of the questions session

References

• Vector potential

• Canonical coordinates

• Gauge invariance

• Lorentz force