13 - Special Relativity, Electrodynamics, Least Action, and Hamiltonians

Abstract

Covered in this chapter are special relativity, electrodynamics, least action principle, Euler–Lagrange equations, and the Lagrangian and Hamiltonian formulations. Topics include Minkowski space and invariant space–time distance, length contraction, and time dilation. Also discussed are Lorentz transformation of space–time position, four-vectors of position and velocity, and four-vector and Lorentz transformation in Minkowski space. Velocity, mass, momentum, and energy are examined in four-vector space. We derived the radiated power from accelerated charge moving at nonrelativistic velocity, using the nonrelativistic Larmor formula and Lorentz transformation applied at each instant of time. Covered next are electromagnetic four-vectors, electromagnetic fields and potentials as four-vectors, the electromagnetic field tensor, the Maxwell stress tensor, and force density. Then, the principle of least action and Lagrangian are used to obtain Euler–Lagrange equations, and the Hamiltonian is derived from the Lagrangian. Next, we derive the Lagrangian for a relativistic charged particle in an electromagnetic field. Finally, we derive the Euler–Lagrange equation for the Langrangian density for real scalar fields.