# Euler-Lagrange equation of a functional

From Diffgeom

## Definition

Let be a differential manifold and be the path space viz the space of piecewise smooth paths from to in . Let be a functional.

The Euler-Lagrange equation is the equation that needs to satisfy to be a critical path of , viz, the equation it must satisfy so that for *every* piecewise smooth variation of , the derivative of along is zero.