If $y \in \mathcal{A}_g$ is a local extremum of $J[y] = \int_a^b L(x, y, y')\, dx$ subject to the holonomic constraint $g(x, y) = 0$, then there exists a multiplier function $\mu: [a,b] \to \mathbb{R}^m$ such that \begin{align*} \frac{\partial L}{\partial y_i} - \frac{d}{dx}\frac{\partial L}{\partial y'_i} + \sum_{j=1}^m \mu_j(x)\, \frac{\partial g_j}{\partial y_i} = 0, \quad i = 1, \ldots, n. \end{align*} Here $\mu_j(x)$ are the **Lagrange multiplier functions** (now functions of $x$, not constants), one for each scalar constraint.