"The functional $J_{\gamma}[y] = \int_{x_0}^{x_1} F(x,y,y')dx, \ y = (y_1,...,y_n),$ where $\gamma$ is a curve described by $y = y(x)$ on $x_0 \leq x \leq x_1$, is said to be invariant under the transformation
$$
x^* = \Phi(x,y_1,...,y_n,y_1',...,y_n') = \Phi(x,y,y'),$$
$$
y_i^* = \Psi_i(x,y_1,...,y_n,y_1',...,y_n') = \Psi_i(x,y,y'),$$
carrying $y = y(x)$ on $x_0 \leq x \leq x_1$ into $y^* = y^*(x^*)$ on $x_0^* \leq x^* \leq x_1^*$, when $J_{\gamma^*}[y^*] = J_{\gamma}[y]$, i.e.
$$
\int_{x_0^*}^{x_1^*} F(x^*,y^*,y'^*)dx^* = \int_{x_0}^{x_1} F(x,y,y')dx."$$