Legendre–Clebsch condition

In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum.

For the problem of minimizing

${\displaystyle \int _{a}^{b}L(t,x,x')\,dt.\,}$

the condition is

${\displaystyle L_{x'x'}(t,x(t),x'(t))\geq 0,\,\forall t\in [a,b]}$