Covariant formulation of classical electrodynamics

Present a covariant relativistic Lagrangian for a charged particle in an electromagnetic field, deriving an invariant action and equations of motion valid at arbitrarily high velocities using the four-vector potential.

Derive two Maxwell equations by applying divergence and curl to the field relations, showing divergence of magnetic field is zero and curl of E equals minus dB/dt (Faraday's law).

Explore the covariant form of the Lorentz force, expressed through the four-velocity and covariant derivative in curved spacetime, with the electromagnetic tensor shaping motion.

Explore the electromagnetic tensor as a 4x4 matrix, relate its entries to electric and magnetic fields, and discuss covariant Maxwell equations and unit conventions.

Rewrite the first two Maxwell's equations in covariant form using the electromagnetic tensor and covariant derivative to preserve the equations' structure across reference frames.

Define alpha nu by dividing by c and examine Maxwell equations' invariance. Derive the electromagnetic energy momentum tensor from a lagrangian density using general relativity and a classical approach.

Differentiate the metric determinant to derive the covariant electromagnetic energy-momentum tensor. Apply the chain rule, explore derivatives with respect to G_AlphaBeta, and use diagonal metric simplifications in a local frame.

Analyze a one-component electric field to compute a piece of the electromagnetic energy-momentum tensor, using Lorentz transformations and F^{μν}, and relate the energy density to ε0 and c.

Explore the distinction between lagrangian density and lagrangian, derive Maxwell's equations from an action principle, and connect four-volume integrals to invariant formulations in covariant electrodynamics.

Derive the remaining Maxwell equations for a general multi-particle or continuous charge system by varying the Lagrangian density with respect to the four-potential, linking current density to the field tensor.

Derive the remaining Maxwell equation in covariant electrodynamics by differentiating the field tensor and assembling terms, showing the four equations are Maxwell's equations.

Show how the covariant derivation of Maxwell's equations arises from a modified Lagrangian, using F^{mu nu} and index raising to obtain divergence and curl relations.

Explore how to derive the energy-momentum tensor from the lagrangian for electromagnetic fields in covariant form, connect to hamiltonian energy, and verify consistency with general relativity definitions.

Explore deriving the electromagnetic energy-momentum tensor from the Lagrangian, including symmetrization and the role of F_alpha beta, and show its divergence-free property using classical Maxwell equations.

Explore the most general energy-momentum tensor from the matter Lagrangian in curved spacetime, via metric variations and four-volume invariance, with Maxwell fields and beyond.

Explore Maxwell's equations to connect the Lorentz force with current density, derive the kinetic energy density rate as E·J, and outline total energy conservation for a closed charge system.

Rewrite Maxwell's equations in index notation to derive the energy momentum tensor and reveal energy conservation through the divergence of the energy flux vector.

Derive the energy-momentum tensor from Maxwell's equations by rewriting the Lorentz force density, substituting curl B and E, and expressing the result in index notation with Levi-Civita symbols.

Derive the electromagnetic energy momentum tensor from Maxwell's equations in covariant form, explain its components and divergence properties, and identify it as the energy momentum tensor.

Explore the covariant formulation of classical electrodynamics, deriving the Maxwell stress tensor and energy density from Maxwell's equations in Minkowski space.

Derive the Lorentz force from the one-particle action and present the first Maxwell equations, showing E from the four-potential and B as curl A.

Derive the second pair of Maxwell's equations from a multi-particle action, introducing the four-current J^μ and the field tensor f_{μν} in Gaussian units.

Derive inhomogeneous wave equations for the four-potential under the Lorenz gauge and solve them by combining homogeneous solutions with a particular integral built from charge elements and Dirac delta sources.

Solve the homogeneous wave equation with spherical symmetry, derive the general solution chi(r,t)=F1(t−r/c)+F2(t+r/c), showing one wave propagates away from the origin and the other toward it.

Derive the retarded potentials for moving charges by solving the wave equation, showing phi and A depend on rho and J at retarded time t−r/c, with near-origin Coulomb behavior.

Learn how the covariant formulation of classical electrodynamics uses potential expansions and a gauge transformation to derive the self-force acting on a moving charge from its own field.

Derive the fundamental equations of geometrical optics from the wave equation, introducing the eikonal function and ray directions via phase gradients.

Investigate image formation in geometrical optics for axially symmetric systems using narrow ray bundles, principal foci and principal points, and analyze lens and mirror cases with magnifications.