Definition:Invariant Functional under Transformation

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

Let $ \mathbf y= \langle y_i \rangle_{1 \le i \le n}$.

Let $ \displaystyle J[ \mathbf y]= \int_{x_0}^{x_1} F \left({ x, \mathbf y, \mathbf y' } \right) \mathrm d x$ be a functional.

Let


 * $ X= \Phi \left({ x, \mathbf y, \mathbf y' } \right)$


 * $ \mathbf Y= \mathbf \Psi \left({ x, \mathbf y, \mathbf y' } \right)$

Let curve $ \gamma$ defined by
 * $ \mathbf y= \mathbf y \left({ x } \right), \quad x_0 \le x \le x_1$

be transformed into a curve $ \Gamma$ defined by


 * $ \mathbf Y= \mathbf Y \left({ X } \right), \quad X_0 \le X \le X_1$

Then the functional is invariant under the given transformation if


 * $J[ \Gamma]=J[ \gamma]$

In other words,


 * $ \displaystyle \int_{x_0}^{x_1} F \left({ x, \mathbf y, \mathbf y' } \right) \mathrm d x = \int_{X_0}^{X_1} F \left({ X, \mathbf Y, \mathbf Y'  } \right) \mathrm d X$