I am trying to find who first translated Maxwell's equations and Lorentz's force into the Lagrangian formalism. It seems a very straightforward thing to do if you know enough of electromagnetism and Lagrangians so maybe nobody cared to give too much credit to the scientist who first published it.
Maybe there were some attemps to do it even before Maxwell's equation?
Who was the first to formulate the Lagrangian of electromagnetism?
In his book on Electrodynamics (Lectures on Theoretical Physics, Vol. 3), Chapter 32-E, Arnold Sommerfeld talks about "Schwarzschild's principle of least action", developed by K. Schwarzschild (Göttinger Nachrichten 1903, English translation here). Recall that the action is just the space-time integral of the Lagrangian density. The principle of least action then varies the action to arrive at the Euler-Lagrange equations. For Schwarzschild's action these are indeed Maxwell's equations and the Lorentz force equation. In a footnote, Sommerfeld says admiringly:
Note the publication date of 1903! Thus Schwarzschild arrived intuitively at the correct postulate of the theory of invariants six years ahead of Minkowski.
Of course, Schwarzschild's Lagrangian also predates Einstein's special relativity by two years. It is indeed remarkable that Schwarzschild obtained the correct Lorentz invariant Lagrangian of electromagnetism at such an early stage.
Schwarzschild himself refers to Lorentz (I believe to this paper from 1892) and Helmholtz (presumably this publication of the same year). Both authors have formulated somewhat clumsy and incomplete precursors to Schwarzschild's (basically) modern version.
Various versions of electromagnetic Lagrangian were "postulated" by elastic aether theorists before Maxwell, see Whittaker, History of the theories of aether and electricity. Green appears to be the first in 1837 (p.151):
"The needful reformation of the elastic-solid theory of reflexion was effected by Green, in a paper read to the Cambridge Philosophical Society in December, 1837. Green, though inferior to Cauchy as an analyst, was his superior in physical insight; instead of designing boundary-equations for the express purpose of yielding Fresnel's sine and tangent formulae, he set to work to determine the conditions which are actually satisfied at the interfaces of real elastic solids. These he obtained by means of general dynamical principles."
Love's Treatise on the mathematical theory of elasticity credits Green with introducing the variational method even into the elasticity theory itself. MacCullagh in 1839 (p.155) and Stokes c. 1867 (p.172) followed him with elastic aether. By Maxwell's time, it was routine. Whittaker mentions that Maxwell himself did it by analogy with elastic theories (p.308), as did Fitzgerald in 1879 with a mistake corrected by Larmor in 1893 (p.368). Lorentz used a Lagrangian to derive the formula for the force later named after him in Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern (1895).
