Suppose $J$ is invariant under the one-parameter group with generator $(\xi, \eta)$. Then the Lagrangian $L$ satisfies the identity \begin{align*} \xi \frac{\partial L}{\partial x} + \eta \cdot \frac{\partial L}{\partial y} + \left(\frac{d\eta}{dx} - y' \frac{d\xi}{dx}\right) \cdot \frac{\partial L}{\partial y'} + L\frac{d\xi}{dx} = 0 \end{align*} along every admissible curve $y$, where $\frac{d}{dx}$ denotes the total derivative along $y$.