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  "body": "\n## Summary\n\nCarathéodory's royal road is a method to prove that a curve minimises an integral functional of the [[calculus of variations|PQ25NHV0KS]].\nOne subtracts from the Lagrangian the total derivative of a function $S(t, x)$, which does not change the problem for curves with fixed endpoints.\nIf $S$ can be chosen so that the new Lagrangian is nonnegative and vanishes along the candidate curve, the candidate is a minimiser.\nThe function $S$ then solves the [[Hamilton–Jacobi equation|5TRXQKP7DY]], so the method links sufficient conditions, [[fields of extremals|K4DFT3H9BJ]] and Hamilton–Jacobi theory.\n\n## Explanations\n\n### L1 human\n\nAdding the derivative of a function to the integrand changes the value of the integral only through the endpoints.\nSo one may look for such a change that makes the integrand nonnegative everywhere and zero along a chosen curve.\nThen no curve can do better than that one.\nFor 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 $\\frac{1}{2} (\\dot x - 1)^2$.\nThis is nonnegative and vanishes only for $\\dot x = 1$, so the straight line $x(t) = t$ is the minimiser.\n",
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