Carathéodory's royal road

Summary

Carathéodory's royal road is a method to prove that a curve minimises an integral functional of the calculus of variations. One subtracts from the Lagrangian the total derivative of a function $S(t, x)$, which does not change the problem for curves with fixed endpoints. If $S$ can be chosen so that the new Lagrangian is nonnegative and vanishes along the candidate curve, the candidate is a minimiser. The function $S$ then solves the Hamilton–Jacobi equation, so the method links sufficient conditions, fields of extremals and Hamilton–Jacobi theory.