---
tag: 4HZX2EN319
kind: statement
role: theorem
name: Legendre condition
msc: [49K05]
uses: [JRQZ803Q5Y, WMPCR723MJ, 4C3BANJ8AH, 01MMVW8CKC]
about: [4C3BANJ8AH]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre%E2%80%93Clebsch_condition
---

## Statement

Let $a < b$, and let $L \colon [a, b] \times \R^n \times \R^n \to \R$, written $L(x, y, p)$, be of class $C^2$.
Let $J(y) = \int_a^b L(x, y(x), y'(x)) \, dx$ be the [[integral functional|JRQZ803Q5Y]] on $C^1([a, b]; \R^n)$.
Let $y_0 \in C^1([a, b]; \R^n)$ be a [[weak local minimiser|WMPCR723MJ]] of $J$ among functions with the same values at $a$ and $b$.
Then for every $x \in [a, b]$ the $n \times n$ matrix $L_{pp}(x, y_0(x), y_0'(x))$ of second derivatives in $p$ is [[positive semidefinite|01MMVW8CKC]].
For $n = 1$ this reads $L_{pp}(x, y_0(x), y_0'(x)) \ge 0$.

## Remarks

The [[second variation|4C3BANJ8AH]] at a minimiser is nonnegative: $\int_a^b \big( h'^T L_{pp} h' + 2 h^T L_{yp} h' + h^T L_{yy} h \big) \, dx \ge 0$ for all $h \in C^1$ with $h(a) = h(b) = 0$.
If $L_{pp}$ had a negative direction $v$ at some interior point $x_0$, take $h$ to be a small zigzag in direction $v$ near $x_0$, with slopes $\pm 1$ and tiny height.
Then the term with $h'^2$ dominates the others, because $h$ itself is small, and the second variation becomes negative.
The endpoints follow by continuity.

Example: for $J(y) = \int_0^1 (1 - y'^2) \, dx$ one has $L_{pp} = -2 < 0$ everywhere, so $J$ has no weak local minimiser with fixed end values.

For a maximiser the matrix is negative semidefinite.
The condition is only necessary: the extremal $y_0 = 0$ of $\int_0^{2\pi} (y'^2 - y^2) \, dx$ satisfies $L_{pp} = 2 > 0$ but is not a minimiser.
The *strengthened Legendre condition* asks that $L_{pp}$ be positive definite; together with the Jacobi condition it gives a sufficient condition for a weak minimum.

## Explanations

### L1 human

At a minimum of an ordinary function, the second derivative cannot be negative.
The Legendre condition is the analogue for curves that minimise an integral.
It says that the integrand must curve upward as a function of the slope, along the minimising curve.

### L2 human

The Legendre condition is the second necessary condition of the classical theory, after the Euler–Lagrange equation.
It only involves the highest-order part of the second variation, the coefficient of $h'^2$.
The full sign of the second variation is controlled by the Jacobi equation and conjugate points.
In several variables its analogue is the Legendre–Hadamard condition, and in optimal control it becomes the Legendre–Clebsch condition on the Hamiltonian.