In a first semester calculus course you learn the derivative can be used to find minimums and maximums of functions, both global and local. Variational calculus lets us extend the idea of criticality to functions satisfying given constraints/functionals, and boundary conditions. The Lagrangian formalism and the principle of stationary action from physics make heavy use of these optimization techniques. Stationary action gives a tool to describe systems from projectile motion and electrodynamics all the way through SU(3) gauge symmetries holding the standard model together. Here I will derive the Euler-Lagrange equations, establish the principle of stationary action, and work through an example of their utility.
