Definition:Invariant Functional under Transformation

Definition
Let $y_i$, $F$, $\Phi$, $\Psi$ be real functions.

Let $\mathbf y=\sequence{y_i}_{1 \le i \le n}$.

Let $\displaystyle J\sqbrk{\mathbf y}=\int_{x_0}^{x_1} \map F {x,\mathbf y,\mathbf y'}\rd x$ 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 is invariant under the given transformation if


 * $J\sqbrk{\Gamma}=J\sqbrk{\gamma}$

In other words,


 * $\displaystyle\int_{x_0}^{x_1} \map F {x,\mathbf y,\mathbf y'}\rd x=\int_{X_0}^{X_1} \map F {X,\mathbf Y,\mathbf Y'}\rd X$