Euler-Lagrange Equation

Euler-Lagrange Equation

I'm starting to get annoyed by that graphic purporting to "explain" physics equations (you know the one I mean). So I figured I'd make my own.

This is the Euler-Lagrange equation, one of the most important equations in classical mechanics and physics in general. It was discovered by Leonhard Euler and Joseph-Louis Lagrange who are rockstars in the mathematical world. It was discovered while attempting to solve the Tautochrone problem: can you find a curve for which the time it takes for a bead to reach the bottom of the curve is independent of starting position. This is a deceptively simple problem which essentially birthed the mathematical field called "variational calculus".

L is called the Lagrangian. When discussing the motion of a particle, it may represent the kinetic energy of the particle minus its potential energy. It is a function of t (you may take this to be time), f (you may take this to be the "y-coordinate" of the particle), and f' (the time derivative of f).

Plugging the Lagrangian into this equation produces a differential equation. The f which solves this differential equation is (with our chosen meaning of f) the "y-coordinate" of the particle as a function of time.

The power of this equation comes from the fact that f does not need to be the "y-coordinate". It can be what is called a "generalized coordinate". There will be a Euler-Lagrange equation for each generalized coordinate. This gives the physicist immense freedom to choose coordinates and thus simplify the resulting equations.

The application of this equation actually results in a whole formalism of classical mechanics called Lagrangian Mechanics. While it's technically equivalent to Newtonian Mechanics, it can produce equations immensely easier to solve. This equation sees application in pretty much every area of physics from electrodynamics to quantum field theory.