\begin{definition}[Definition:Invariant Functional under Transformation]
Let $y_i$, $F$, $\Phi$, $\Psi$ be real functions.
Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$.
Let $\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional.
Let:
:$X = \map \Phi {x, \mathbf y, \mathbf y'}$
:$\mathbf Y = \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'}$
Let curve $\gamma$ defined by: 
:$\mathbf y = \map {\mathbf y} x, \quad x_0 \le x \le x_1$ 
be transformed into a curve $\Gamma$ defined by:
:$\mathbf Y = \map {\mathbf Y} X, \quad X_0 \le X \le X_1$
Then the functional $J \sqbrk {\mathbf y}$ is '''invariant''' under the given transformation {{iff}}:
:$J \sqbrk \Gamma = J \sqbrk \gamma$
That is, {{iff}}:
:$\ds \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{X_0}^{X_1} \map F {X, \mathbf Y, \mathbf Y'} \rd X$
\end{definition}