Euler-Lagrange equation of a functional

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Definition

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.