# Euler-Lagrange equation of a functional

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Let $M$ be a differential manifold and $\Omega(M;p,q)$ be the path space viz the space of piecewise smooth paths from $p$ to $q$ in $M$. Let $F:\Omega \to \R$ be a functional.
The Euler-Lagrange equation is the equation that $\omega \in \Omega$ 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.