Let $y \in \mathcal{A}_\ell$ be a local extremum of $J[y]$ subject to the constraint $K[y] = \ell$. Suppose $y$ is not an extremal of $K[y]$ alone (the **abnormality condition**). Then there exists a constant $\lambda \in \mathbb{R}$ (the **Lagrange multiplier**) such that $y$ satisfies the Euler–Lagrange equation for the augmented Lagrangian $\tilde{L} = L - \lambda M$: \begin{align*} \frac{\partial (L - \lambda M)}{\partial y} - \frac{d}{dx}\frac{\partial (L - \lambda M)}{\partial y'} = 0. \end{align*}