Let $L \in C^2$ and let $\Phi_s$ be a one-parameter group of transformations with generator $(\xi, \eta)$ under which $J[y] = \int_a^b L(x,y,y')\,dx$ is invariant. If $y$ is a solution of the Euler–Lagrange equations \begin{align*} \frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} = 0, \end{align*} then the quantity \begin{align*} I(x, y, y') := \frac{\partial L}{\partial y'} \cdot (\eta - y'\xi) + L\xi \end{align*} is constant along $y$, that is, $\frac{d}{dx} I(x, y(x), y'(x)) = 0$.