Tensor Lagrangians, Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics


We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell’s equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.

This is a preview of subscription content, log in to check access.


  1. 1.

    Gersten, A.: Maxwell equations as the one-photon quantum equation. Found. Phys. Lett. 12, 291–298 (1999)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Gersten, A.: Maxwell equations—the one-photon quantum equation. Found. Phys. 13, 1211–1231 (2001)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Bialynicki-Birula, I.: Photon wave function. Prog. Opt. 36, 245 (1996)

    Article  Google Scholar 

  4. 4.

    Esposito, S.: Found. Phys. 28, 231–244 (1999)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Gersten, A.: Quantum equations for massless particles of any spin. Found. Phys. Lett. 13, 185–192 (2000)

    Article  MathSciNet  Google Scholar 

  6. 6.

    Fushchich, V.I., Nikitin, A.G.: Symmetries of Maxwell’s Equations. Reidel, Dordrecht (1987)

    Google Scholar 

  7. 7.

    Gersten, A.: Conserved currents of the Maxwell equations with electric and magnetic sources. Ann. Fond. Louis de Broglie 21, 67–79 (1996)

    Google Scholar 

  8. 8.

    Morgan, T.A., Joseph, D.W.: Tensor Lagrangians and generalized conservation laws for free fields. Nuovo Cimento 39, 494–503 (1965)

    Article  MathSciNet  Google Scholar 

  9. 9.

    Sudbery, A.: A vector Lagrangian for the electromagnetic field. J. Phys. A, Math. Gen. 19, L33–L36 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  10. 10.

    Fushchych, W.I., Krivskiy, I.Yu., Simulik, V.M.: On vector and pseudovector Lagrangians for electromagnetic field. Nuovo Cimento B 103, 423–429 (1989). Also in W.I. Fushchych, Scientific Works 3, 506–510 (2001)

    Article  ADS  Google Scholar 

  11. 11.

    Gersten, A.: Field approach to classical mechanics. Found. Phys. 35, 1433–1443 (2005)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  12. 12.

    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading (1980)

    Google Scholar 

  13. 13.

    Pauli, W., Fierz, M.: Proc. R. Soc. Lond. A 73, 211 (1939)

    MathSciNet  Google Scholar 

  14. 14.

    Corson, E.M.: Introduction to Tensors, Spinors and Relativistic Wave-Equations. Hafner, New York (1953)

    Google Scholar 

  15. 15.

    Barut, A.O.: Electrodynamics and Classical Theory of Fields and Particles. Dover, New York (1980)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Alexander Gersten.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gersten, A. Tensor Lagrangians, Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics. Found Phys 41, 88–98 (2011). https://doi.org/10.1007/s10701-009-9352-3

Download citation


  • Scalar Lagrangians
  • Tensor Lagrangians
  • Hamilton-Jacobi equation
  • Relativistic dynamics