Euler-Lagrange equation of a functional

Definition

Let $M$ be a differential manifold and $\mathrm{\Omega }(M;p,q)$ be the path space viz the space of piecewise smooth paths from $p$ to $q$ in $M$. Let $F:\mathrm{\Omega }\to \mathbb{R}$ be a functional.

The Euler-Lagrange equation is the equation that $\omega \in \mathrm{\Omega }$ needs to satisfy to be a critical path of $F$, viz, the equation it must satisfy so that for every piecewise smooth variation $W$ of $\omega$, the derivative of $F$ along $W$ is zero.