Editing Karl Schwarzschild (section)

===Electrodynamics===

According to [[Wolfgang Pauli]],<ref>Pauli, W.. Theory of Relativity. United States, Dover Publications, 2013.</ref> Schwarzschild is the first to introduce the correct [[Lagrangian (field theory)|Lagrangian]] formalism of the electromagnetic field<ref name= "schwarzschil1903">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 125</ref> as

:<math> S = \frac12 \int (H^2-E^2) \mathrm dV + \int \rho(\phi - \mathbf{A} \cdot \mathbf{u}) \mathrm dV </math>

where <math> \mathbf{E},\mathbf{H} </math> are the electric and applied magnetic fields, <math>\mathbf{A}</math> is the vector potential and <math>\phi</math> is the electric potential.

He also introduced a field free variational formulation of electrodynamics (also known as "action at distance" or "direct interparticle action") based only on the world line of particles as<ref name= "schwarzschil1903b">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 128,132</ref>

:<math>
S=\sum_{i}m_{i}\int_{C_{i}}\mathrm ds_{i}+\frac{1}{2}\sum_{i,j}\iint_{C_{i},C_{j}}q_{i}q_{j}\delta\left(\left\Vert P_{i}P_{j}\right\Vert \right)\mathrm d\mathbf{s}_{i}\mathrm d\mathbf{s}_{j}
</math>

where <math> C_\alpha </math> are the world lines of the particle, <math> d\mathbf{s}_{\alpha} </math> the (vectorial) arc element along the world line. Two points on two world lines contribute to the Lagrangian (are coupled) only if they are a zero Minkowskian distance (connected by a light ray), hence the term <math> \delta\left(\left\Vert P_{i}P_{j}\right\Vert \right) </math>. The idea was further developed by [[Hugo Tetrode]]<ref>H. Tetrode, Zeitschrift für Physik 10:137, 1922</ref>
and [[Adriaan Fokker]]<ref>A. D. Fokker, Zeitschrift für Physik 58:386, 1929</ref> in the 1920s and [[John Archibald Wheeler]] and [[Richard Feynman]] in the 1940s<ref>{{Cite journal |last1=Wheeler |first1=John Archibald |last2=Feynman |first2=Richard Phillips |date=1949-07-01 |title=Classical Electrodynamics in Terms of Direct Interparticle Action |journal=Reviews of Modern Physics |language=en |volume=21 |issue=3 |pages=425–433 |doi=10.1103/RevModPhys.21.425 |issn=0034-6861|doi-access=free |bibcode=1949RvMP...21..425W }}</ref> and constitutes an alternative but equivalent formulation of electrodynamics.