Euler-Lagrange equation

The Euler-Lagrange equation of a functional $L(u)$ is an equation which is necessarily satisfied (formally, at least) by critical points of that functional. It can be computed formally by starting with the equation

${\frac {d}{d\epsilon }}L(u+\epsilon v)|_{\epsilon =0}=0$

for arbitrary test functions v, and then using duality to eliminate v.

Equations which are Hamiltonian can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle be reformulated in a Hamiltonian manner.