Derivative of functional on path space

Definition

Let $M$ be a differential manifold and $\Omega =\Omega (M;p,q)$ be a path space on $M$ (viz, the space of piecewise smooth paths from $p$ to $q$ in $M$.

Let $W$ be an element of the tangent space at a path $\omega \in \Omega$ and let $F$ be a functional from $\Omega$ to $\mathbb {R}$. Then the derivative of $F$ at the point $\omega \in \Omega$ with respect to the tangent direction $W$ is defined as follows:

Let $\alpha$ be any variation whose variation vector field is $W$. The derivative is then:

${\frac {d}{du}}|_{u=0}F(\alpha (u))$

A critical path of a functional is defined as a path (element of the path space) at which the derivative of the functional is zero.