Euler operator

Also Euler-Lagrange operator.
It is an operator acting on functions with domain in the jet bundle $J^{n}$ (differential functions). In the case of $p = q = 1$ is given by

\sum_{j = 0}^{n} {(- D_{x})}^{j} \frac{\partial}{\partial u_{j}}

being $D_{x}$ the total derivative operator.
For the motivation for this definition see variational derivative#Some facts.

In the general case is defined as

E_{α} = \sum_{J} (- D)_{J} \frac{\partial}{\partial u_{J}^{α}}

(see @olver86 page 246).

Euler operator gives 0 when applied to a function $L$ coming from a total divergence. This is shown in @olver86. Is part of a complex, the variational bicomplex with the operators: $Div$ and Helmholtz operator. See my own question in MO.

I think it appeared in the context of the inverse problem for Lagrangian mechanics.