What is the proof of the principle of least action?

Answer 1

The Principle of Least Action (more correctly Stationary Action as it is not necessarily a minimum) is derived from the Euler Lagrange equations. What confuses people and makes it seem like a mysterious assumption is that a "classic" textbook question asks one to show that given the Principle of Least Action show that the Euler-Lagrange equations must hold. This leads to the misconception that the derivation is the other way i.e. that the Euler-Lagrange equations are derived from this unexplained Principle of Least Action. The standard Euler-Lagrange equations with a zero on one side are derived from Newtons Laws and are only true when forces derived from a potential energy are considered, the equation is non-zero if other forces are present and in such cases the Principle of Least Action doesn't hold (those other forces cause the action to deviate from the stationary point value). Taking this into consideration, the Principle of Least Action is simply a disguised version of the trivially true statement that a system in unperturbed if there are no perturbations present - or in more Newtonian terms, things won't accelerate unless there is an outside force acting on them.