All the fundamental laws of physics can be written in terms of an action principle. This includes electromagnetism, general relativity, the standard model of particle physics, and attempts to go beyond the known laws of physics such as string theory. For example, (nearly) everything we know about the universe is captured in a Lagrangian where the terms carry the contributions of Einstein, Maxwell (or Yang and Mills) and Dirac respectively, and describe gravity, the forces of nature (electromagnetism and the nuclear forces) and the dynamics of particles like electrons and quarks.
Source: David Tong
Me: I am a second year undergrad and have a nice familiarity with the -1/4 F_ij F_ij term (electromagnetism) and how it results to the Maxwell's Equations, but am still curious to know how it seems like there is always a Lagrangian. Apart from one obvious advantage that the Euler Lagrange equations hold in any coordinate system and the Lagrangian holds the key to the symmetries of the system.