Derivative of functional on path space

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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 \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.