Deriving the Lagrangian Density of an Electromagnetic Field

  • Bahman Zohuri


As part of understanding relativistic particles and electromagnetic fields, we need to have some awareness of the kinematic special theory of relativity, and then turn our attention to the dynamic aspect of charged particle motion in external electromagnetic fields. Thus, in this chapter we take a Lagrangian approach to the equations of motion and deal with the total energy involved with the motion of a particle. We also introduce the Hamiltonian in relation to the total energy of particle motion; analogous to classical mechanics, it relates to the corresponding kinetic and potential energy of the particle or system of concern. The Lagrangian approach is based in electrodynamics; thus, equations of motion are presented mainly as an avenue to introduce the concept of a Lorentz invariant action to a Hamiltonian, with the definition of the canonical momentum discussed in this chapter.


  1. 1.
    F.C.G. Stueckelberg, Interaction forces in electrodynamics and in field theory of nuclear forces. Helv. Phys. Acta. 11, 299–328 (1938)Google Scholar
  2. 2.
    D. Griffiths, Introduction to electrodynamics, 3rd edn. (Prentice Hall, Upper Saddle River, 1999)Google Scholar
  3. 3.
    C.K. Law, Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. Phys. Rev. A 51, 2537 (1995)CrossRefGoogle Scholar
  4. 4.
    D.A. Woodside, Three-vector and scalar field identities and uniqueness theorems in Euclidean and Minkowski spaces. Am. J. Phys. 77, 438–446 (2009)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Bahman Zohuri
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of New Mexico, Galaxy Advanced Engineering, Inc.AlbuquerqueUSA

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