---
tag: 148S3XETT3
kind: concept
name: Carathéodory's royal road
msc: [49K05, 49L12]
sources:
  - url: https://en.wikipedia.org/wiki/Calculus_of_variations
---

## Summary

Carathéodory's royal road is a method to prove that a curve minimises an integral functional of the [[calculus of variations|PQ25NHV0KS]].
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|5TRXQKP7DY]], so the method links sufficient conditions, [[fields of extremals|K4DFT3H9BJ]] and Hamilton–Jacobi theory.

## Explanations

### L1 human

Adding the derivative of a function to the integrand changes the value of the integral only through the endpoints.
So one may look for such a change that makes the integrand nonnegative everywhere and zero along a chosen curve.
Then no curve can do better than that one.
For example, for $\int_0^1 \frac{1}{2} \dot x^2 \, dt$ with $x(0) = 0$ and $x(1) = 1$, subtracting the derivative of $x - t/2$ turns the integrand into $\frac{1}{2} (\dot x - 1)^2$.
This is nonnegative and vanishes only for $\dot x = 1$, so the straight line $x(t) = t$ is the minimiser.